Well, either the altitude info is coming from a geospatial database that has the elevations at close enough accuracy and 2D frequency to map usefully to your recorded GPS waypoints or you need an altitude accurate device to take with you to each location.

I don't know if there is a sufficiently accurate ground profile database for your area, nor have I any idea how one would get access to one if it exists.

Consumer handheld GPS is not intended to provide high degrees of accuracy for altitude. The primary focus is on x-y accuracy and speed of determining current location. I think there are some high accuracy handheld GPS products that have better altitude accuracy.

Perhaps the GPS products sold for land survey work would provide the necessary accuracy. No idea on cost or how usable such a unit might be for your purposes. This land survey model is apparently accurate to 2cm vertical.