I usually realize he has said something to upset me after the session too. I have to just sit with it until the next session the following week which is really really difficult as the feelings tend to get worse and more exaggerated despite my best efforts during that time.

I think it would be much better and helpful if I could email him my thoughts/hurt/anger as soon as I realize even if he was just to reply with an acknowledgment of receiving the email and potential to discuss next session. At the start, however, he did say he doesn't 'do' email unless it's a scheduling issue.

It seems somehow cruel to have to sit with such terrible (and often unnecessary) hurt and pain. Usually, it ends up being a misunderstanding or misinterpretation of something he said, me overreacting to something and realizing it within seconds or he apologizes. If a friend or even just a colleague (which I know a T is neither) said something to upset me I think I would bring it up with them as soon as possible so as not to drive myself crazy dwelling on it.
When I do finally get to share the upset with my T in the next session it dissipates within minutes of discussion so it would be helpful to have this sooner IMO.