(This is an edited and expanded answer of mine to a previous question about the later history of the military orders.)

Firstly, your question implies that there was an end to the crusades sometime before the Hospitallers conquered Rhodes (around 1310). When exactly the crusades ended is a matter of some debate amongst historians (as is what should be considered a crusade). However, most historians now accept that crusading did not end with the fall of Acre in 1291, or the loss of Ruad in 1302, and the final expulsion of the Crusader States from Palestine.

There were quite a few crusades in the 14th century and for some countries, like Scotland and England, this period was one of the most popular for crusading. However, these crusades were very different to those of the 12th and 13th centuries. The major crusades then, like the Third Crusade, were mass expeditions launched by the Pope and often led by kings. With the loss of the Holy Land in 1291, when Acre was conquered by the Saracens, these mass royal crusades ended. There was no longer a foothold in Palestine for the crusaders to strike out from. Crusading to the East became more risky and expensive.

In response, crusading rose in popularity elsewhere. Crusaders began to travel throughout the hethenesse, a Middle English word that means the non-Christian borders of Europe and Africa. The hethenesse included Spain, where the Christian Spanish kingdoms had been slowly beating back the Muslim Moors for centuries, the Baltic, where the Teutonic Knights were fighting pagan Lithuania, Greece and the eastern Mediterranean, where the Knights Hospitaller and the Byzantine Empire were fighting the Turks, and North Africa.

Crusading also continued into the fifteenth century, ending in the Baltic after the Teutonic Order's disastrous defeat at the battle of Tannenburg/Grunwald in 1410. In the Mediterranean the Hospitallers and other minor military orders continued to attack Turkish and Barbary shipping until their expulsion from Malta in 1798. Some historians have also traced the crusading ideology in Spain and Portugal's early conquests of Africa and the Americas. Many of the commanders of these expeditions were members of Iberian military orders like the Knights of Christ. If you want to look into late medieval and early modern crusading more, then Norman Housley's The Later Crusades is a good option, it goes from about 1270-1580.

As to the military orders in the late medieval period, the Spanish military orders, the Knights of Santiago, Order of Calatrava, Order of Alcantara, and others, secularised. Throughout the late medieval period they increasingly served as royal commanders and also participated heavily in Portugal's first expeditions along the West African coast. In 1523 all the Spanish orders were taken under royal control, they abandoned their monastic vows and lifestyle, and eventually became the monarchical Spanish orders they are today, and the same process happened to the Portuguese military orders.

The Hospitallers are still around today, as the Sovereign Military Order of Malta (if you can prove that your recent ancestors were nobles, that you're a practicing Catholic, and that you're a genuine supporter of the Order then you could join today as a Knight/Dame of Justice). The Order's later history was a bit more complicated than that of the Spanish orders. In England, Ireland, Wales, and Scotland they were suppressed at the Reformation. In Germany the Bailliwick (an administrative division of the Hospitallers) of Brandenburg broke away and converted to Lutheranism. But the bulk of the Order, now based on Malta, remained Catholic and conducted pirate hunting campaigns (and some piracy of their own) against Muslim ships in the Mediterranean. They retained their lands in France, the Iberian Peninsula, Italy, and much of Germany. They even bought a few Caribbean colonies, including St Barthelemy, from France in the 1650s, but sold them soon after. The Hospitallers' naval success became quite famous. From the mid 1600s until 1748 the Order trained and commanded many of the French navy's galleys. From the 1760s until 1796, the Hospitallers also hosted and trained Russian naval officers. The Order continued these naval campaigns all the way up to 1798 when Napoleon conquered Malta. After this some of the Order relocated to Russia, which had offered them sanctuary. The Tsar was appointed grandmaster (despite not being Catholic) Eventually the Order settled in Rome in 1834, their time in Russia deemed an embarassing episode that should be brushed over. The Order is still headquartered in Rome, though they also now have some presence in Malta again.