My T seems connected and caring during the session but we don't have contact outside of the session. I don't even have his phone number—I schedule appointments through the office. For all I know he only thinks about me for those 50 minutes a week + however long it takes him to do my paperwork.
There was one time I got a second appointment on short notice because I was in a mini crisis and I called his office to see if they had any last minute openings and there had been a cancellation. But even that was just lucky scheduling through the office, not him going out of his way to make extra time for me.

I don't think this precludes a caring therapeutic relationship. For me it's a form of boundaries. It keeps me from acting out to get extra attention between sessions because I know I wouldn't get it. It's easier not to have the temptation of being able to reach out to him or to push the boundaries just a little bit more. I'm sure if I was in the hospital or something they could contact him, but other than that I just have to be patient (which is certainly not easy).
I've had therapeutic relationships that were too close in the past. At the time it made me feel special, but it wasn't as helpful and it kept me from getting real work done.
It was just enabling me and I was recreating the same pattern of trying to get someone to fill the emotional void through enough caring or praise or love, when that isn't possible, and what I need is to work through coming to terms with that and learning to do those things for myself rather than seeking them externally.

I don't know if this is the case for you. Your situation and needs might be totally different. But it's been important for me to realize that what I need might be different from what I want. I think of it as taking a pain pill vs physical therapy. A pain pill makes it feel better, but doesn't actually solve the problem. Physical therapy hurts and takes time but can eventually lead to progress with the underlying problem. And combining pain pills with physical therapy would be a bad idea in this analogy, because I could be creating more damage and not realize it without the pain to tell me. (Okay my analogy admittedly got stretched a little thin there, but the point is still there)

I also want to be able to trust that my therapist is maintaining the boundaries they need. I don't want to have to worry that I could burn them out or take too much. I want to trust that they're looking after their own needs so that they wouldn't let that happen.

Good luck with your therapist. I hope things work out and you're able to discuss this with him and be on the same page.

(Wow this was really long, sorry)