Since your storage conditions are good environmentally, the next thing I’d consider is the risk of physical damage – does the hair need to be removed from the storage occasionally/often? Who will take it out of its packaging?
If the hair is going to stay in its ziplock bag most of the time and will mostly be viewed while still in the bag, then I would be inclined to think bags are a good idea. I would, however, put a small rectangle of acid-free card (slightly smaller than the access to the bag) and place the hair on that. This provides support for the hair, prevents the bag distorting and possibly reshaping the hair, acts as an environmental buffer to stabilize RH even further and also may help absorb any sulphurous compounds given off as the hair ages. It will make it easier to see the hair in the bag and provides a surface which can be labelled/marked (on the reverse).
Small bags can be stood upright if they have card inserts, and kept in a secure box, so that the items take up less space and the hair is not being compressed by having other bags laid on top as it would if they were laid flat in a drawer.
I presume the area is well monitored for pests?
Don’t forget to keep the hair well away from photographs and items containing silver as the sulphur-rich keratin of the hair can cause problems for silver compounds.
Hope this helps.