Hi LT, I totally understand that cycle, and the horrible worries that can come up if you contact T and they don't reply straight away!
With regards to being direct about it, I often was, because my T led the way, and I think it was helpful. T used to say "feel free to contact me if you want to share something, or just to check I'm still there". We also spoke openly in therapy about how sometimes I didn't have a specific problem but I just wanted to touch base with him - he said it was fine, and he was glad it was helpful.
I think this was good because it eliminated the potential "shame" - that I'm an adult and I should be fine and I shouldn't need contact from this person who is not my family, etc. I think it was helpful to be open, and that my T was totally fine about it and even encouraged it, and reassured me that it was a helpful part of therapy.
I still need this contact sometimes, but less often. And I don't get so worried if I don't hear back straight away. So in my experience, it does get easier!