Congrats on starting T in a few weeks, Kade!
I'm not sure whether you would want to come out at work right now even if you weren't going to start T, but you may not
need to come out just because of the effects of T. At least not for the first 3-4 months and possibly longer — even if your voice starts to get deeper and you start looking a little more masculine.
It's not that changes on T are slow—even though sometimes they are—it's that the changes are gradual enough that if someone sees you every day, it'll take them a while to notice that anything out of the ordinary has happened. It's really pretty incredible the sort of things folks will ignore, too. When I started binding years ago, no one said a thing other than "Hey, did you lose weight?"

I've been on T for 4 months and I still haven't gotten any comments about my transition from doctors don't know I'm transitioning or other people I know, other than the occasional "Have you been sick?" because of my voice. Yet perfect strangers regularly call me "Sir".
So, if being seen as female at work doesn't make you want to jump out of your skin, you could probably continue to work there without coming out as a guy. Any changes short of a very nice and luxurious beard could probably be shrugged off as getting older and growing out of the "babyfaced" look everyone has when they're younger.
Unfortunately, TN doesn't have any laws on the books prohibiting employers from discriminating on the basis of sexual orientation or gender identity, so you'll have to rely on the policies the staffing agency has in place. And I agree with the previous posters re: talking to the staffing agency first, rather than the phone company.
Is the staffing agency a national one (Adecco, Manpower) or a smaller regional one? If it's a national one, I suggest getting in touch with their main office to talk with them about their policies re: gender transition on the job. Heck, even if it's a Tennessee-based company, if they have a main office in Nashville then those folks are probably the ones you want to talk to. That way it can be somewhat anonymous (at first)
and you'll know exactly what the company's official stance is before talking to your local office.
(All the big cities in TN
do have anti-discrimination laws that cover transgender folks, btw. Just in case you're thinking of moving somewhere less rural someday.)
Good luck!