The kind where a nation was low on troops so they hired armies to fight for them.

This I am not sure of. There were however private armies in service to merchants (specifically the Aztec oztomeca, the long distance traders that were part of the pochteca merchant class). There are at least a few examples of nobility recruiting these private security forces into their armies.

An example of this can be found in a document known as La Memoría de Don Melchor Caltzín, a 16th century testimony given to a Spanish court in Pátzcuaro, Michoacan, but supposedly taken from a pictographic codex which has not survived. The account describes a group of wealthy Nahuatl-speaking merchants intervening in a succession war between various claimants to the throne of the early Tarascan empire (probably, the document is written in Early Colonial P'urépecha and much of it is obscure). One claimant, Tzitzispandáquare, was attempting to take the capital city of Tzíntzuntzan, which was likely under the control of a rival claimant. He lacked adequate troops, so he appealed to a group of Nahua merchants for assistance. They agreed to lend him soldiers in their employ in exchange for noble titles. Since these merchants were not nobility, and thus did not have any official armies that served them, it is likely these were paid soldiers (i.e., mercenaries).

I'm not sure if we can say whether or not it was common practice for nobles to recruit soldiers in this way to flesh out their armies, but it does suggest that it happened occasionally. At the least, merchants appear to have employed private security to protect them on long trade runs.

A notable example in the Eastern Woodlands would be the relationship between the Powhatan and the Chickahominy. The Chickahominy were something of a client nation surrounded by the Powhatan. This map shows the Chickahominy as the gap within Powhatan weroancy, though it doesn't show the Powhatan at their greatest extent (which would push their border north of the Potomac and across Chesapeake Bay to include Accomac). The Chickahominy maintained their own political system, based around a more egalitarian council of elders known as the mungai rather than a hierarchy of weroances. The weroances are what the English called "kings" or "chiefs." Lesser weroances oversaw individual communities and paid tribute to greater weroances who had regional authority within the weroancy. At the top of the hierarchy was the mamanatowick, Chief Powhatan himself at the time of European contact, from whom authority flowed and to whom all tribute rose at his capital, Werowocomoco.

When the mamanatowick decided it was time to go war, he dispatched orders to the weroances, who then conscripted able-bodied men to form units under their command. The mungai did not possess this sort of authority however and in order to obtain warriors from the Chickahominy, the mamanatowick hired warriors through payments of copper, a resource he monopolized for such purposes. The recruitment of Chickahominy warriors was important in maintaining the peace between the weroances, as they formed a force that the mamanatowick alone could call upon in order to put down rebellions and police inter-weroance rivalries.