I think the bottom line is that yes, there are full-time vet positions in rehab, and yes, there are full-time vet positions in zoo med ... but both those occupations are extremely limited in terms of income and positions available. So you need to go into it with both eyes wide open to that.
If you're really set on it, then you do what people have always done in competitive fields: you be super competitive to give yourself a shot. You start now on the path to accumulating a crap-ton of experience, connections, and knowledge. In the end, the people that love it so much that they're constantly reading about it, volunteering, etc., are going to be the ones that get the good jobs.
If you're only so-so set on it, or you think "gee that would be kinda cool" then it might be best to consider another field, or consider doing it as a volunteer, or whatever. If you want to do it because you think hand-feeding a fawn is neat ... why not just volunteer?
I volunteered in wildlife rehab for a year before vet school at the MN Wildlife Rehab Center. I think we were a little unusual in our size .. we had two full-time vets, two full-time CVTs, externs, volunteers, full-time staff, a hospital with multiple wards, a well-stocked treatment and surgery, etc. It was larger than most small animal clinics. I don't think your typical rehab center is that well-equipped/staffed.
I dunno. It's just such a limited field that I'm hard-pressed to see making it a full-time gig that supports your life unless you're one of those few people who just really, really love it and make it their entire life.