It is 1166 and Tancred of Lecce has just returned to his fief, the County of Puglia, in the Kingdom of Sicily. William, the king who exiled him for leading a sanguine revolt, has died and his twelve-year-old son has ascended to the throne under the tutelage of a regency council. But while the King he had wronged is dead, Tancred still has obstacles he will need to overcome upon his return: he needs to reassert rights and privileges in his own fief, as well as asserting his role within the kingdom. In order to do this, he cannot rely on relationships and institutions that have deteriorated in his absence. He will need to find resources by unconventional means.
Fortunately, Tancred was an enthusiastic navigator with good relationships amongst the small but industrious mercantile class in the city of Lecce. It is this network that would have helped him set up a small household during his exile, which would include not only his innermost circle, but also a small bodyguard of Greek veterans. So the first answer to your question is that men who knew how to handle arms and armor could always be contracted on an individual basis, which for Tancred was made easier thanks to his network of extended friends and acquaintances in Greece (which he did well to strengthen even more upon his arrival).
But now that he was back in Lecce, Tancred's mercantile contacts wouldn't be as useful. He must stabilize the whole of Puglia, and Leccese merchants would have especially weak ties to the countryside where Tancred would theoretically have the strongest rights and derive the majority of his power.
From Lecce, an itinerary across his castles and manors in Puglia reveals the situation is more dire than he thought. Although his castellans, trusted relatives and associates, readily hand over the keys it is clear that they lack the will or prestige to collect tithes from his vavasours. It is these vavasours that he would typically call upon to defend Puglia from Saracen raids, enforce his privileges, as well as protect his claims against rivals and against impositions by the monarchy. However, over the past five years he has been unable to maintain relationships, he has lost track of appointments of bishops and abbots, and has not been informed on appointments to town councils. This means that if Tancred is to return to a position of prominence in the Kingdom of Sicily, he must find a way to return to the top of the social hierarchy he has been cut off from.
This is, in part, to be expected. Tancred chose his castellans for their loyalty, and if they had been able to impose themselves on his vavasours he would have risked creating dangerous rivals who might not look upon his return with enthusiasm. So how could Tancred collect his income and impose authority on his vavasours without pushing them to revolt?
To project power Tancred could solicit the city of Lecce for help. While the landowning and rent-collecting class had the time and resources to train with arms and armor, prosperous urbanites could also be expected to take up arms when their homes and livelihoods were threatened. In large enough cities, mercantile guilds could organize the armament and training of their members to defend the city in times of need, while the crossbow was a fairly popular weapon amongst both the bourgeoisie and the mercantile community, while wealthier burghers could even come to own suburban estates and keep horses.
But the burghers of Lecce won't take up arms to risk life and limb unless they are themselves threatened. Tancred would have to offer something in return. Indeed, in the next decade Tancerd's enthusiastic recruitment of sailors and ships for King William II's campaigns implies an effort to bestow favors and prominence to the maritime communities in Puglia, while a flourishing artistic and architectural scene under his tutelage implies attempts to win the favor of the influential clergy. Winning the favor of the burghers in Otranto, Taranto, and Bari in addition to Lecce, would create some additional revenues, a safe haven within city walls, and perhaps some additional influence in courts of law. However, it would not be enough to convince burghers to take up arms for him. They might protect him against rivals and unruly vavasours, however if faced with concentrated effort to topple him they might just as easily hand him over.
So having secured revenue and a safe haven, if Tancred were to set out to seize the goods and property of a particularly disobedient vavasour in 1166 or 1167, where would he find the manpower?
While it's not accurate to say there was no professional soldiery in Medieval Europe, it is true that their numbers would have been small: household servants trained to protect members of aristocratic families, strongmen contracted to protect trading ships and caravans, and only sometimes gatekeepers and night watchmen (these could just as easily be volunteer positions). The only place to quickly find armed men ready for battle was, perhaps predictably, after a battle.
Tancred was fortunate, as the German Kaiser Frederick Barbarossa was presently seated in Rome, having just scored a resounding victory against Papist forces in the battle of Prataporci. Some Roman aristocrats had locked themselves in their palaces, others fled to the countryside. Still other aristocrats and burghers who had no interest in staying in a Rome occupied by the Emperor found their way to various places in Italy, including Sicily and Puglia.
Among those who left, smaller burghers and landowners expected they would lose more by staying than leaving. Some might have friends, family, or associates with whom they could seek refuge. Others might accept contracts to accompany Palermitan or Leccese merchants and protect their cargo as they fled the inevitable breakdown of law and order. Thus Tancred's associates, friends, and relatives who might be in Palermo, in the Roman harbor of Ostia, or even in the cities of Puglia, whom he had told he was looking to recruit fighting men for the campaigning season of 1167, would know to make introductions.
While legal documents (notably notary records) can trace the journey of some soldiers of fortune, more commonly they just materialize with little explanation (John Hawkwood's company is raiding southern France one season and defending the Marquis of Monferrat the next one; we do not know who made the introduction). But we do know that there were Leccese clergymen and merchants in Rome who would want to flee the conflict, and it is not unreasonable to assume they would approach or be approached by survivors of Prataporici to guard their person and their goods on the return journey to Lecce. Upon arriving in Lecce, the papists might request an audience before a member of the city council (perhaps with introductions helped along by the merchant who contracted services in the first place). The member of the city council, in turn, would be in Tancred's circle and most certainly know he is planning a punitive expedition.
This is to demonstrate how Tancred could expect that the extensive social and economic network of both his social and political circle could reach out to connections in or near conflict zones, where they could communicate that Tancred, the Count of Puglia, would accept pledges of service.
Tancred would soon demonstrate a third way to raise fighting men outside his typical scope of recruitment after he reasserted himself in Puglia. King William II died before his time, and had designated his aunt Constance as successor. But because she was betrothed to Henry Salian, Emperor Frederick Barbarossa's son, the Southern Italian aristocracy was less than enthusiastic about her ascension. While they could hope to exert a fair degree of influence a Sicilian monarch, asserting their privileges would be much more difficult if their King was able to call upon countless German soldiers hungry for glory and plunder. The prospect of a joint German-Sicilian campaign against the increasingly autonomous Northern Italian cities was also concerning, which would pit the Southern Italians against their most important trade partners.
This brings us to the third way a medieval politician could raise fighting men: convince your opponents supporters to defect to your camp. Tancred did exactly this: Constance left Sicily to be wed, and Tancred took the opportunity to convince even her closest vassals in Sicily to support him as he seated himself in Palermo and was proclaimed King.
I have extensively embellished the life of Tancred of Lecce above, but the actual historic events described (Tancred's exile, return, naval career, and ascension) are all factual. The social and economic dynamic of the Southern Italian political system is also factually described. What I'm trying to demonstrate is the way that "unconventional" military recruitment occurred (I have ignored, for example, conflicts Tancred managed after he stabilized Puglia, as well as conflicts against rebellious subjects after his coronation). It is important to understand that this is a time before standing armies (even the wealthiest households' fighting men would have additional responsibilities) and those who were recruited to fight were those who could afford to keep arms and armor. In convincing these people to take up arms, a social and political network would have to be mobilized. Success would depend on the strength of relationships that could be leveraged, in addition to the gold awarded as well as plunder that could be credibly promised. This was even more true when outside of conventional recruitment, which would have been more aligned with institutionalized ties of homage and interdependence.