My experience is just as Ellahmae.
I would never have expected or asked for between session contract when I first began seeing T. I did know that I could leave a message on her office phone in an emergency and get a call back. That was fortunately not ever needed.

But after awhile, she offered me her personal cell and said that she'd actually like for me to text her between sessions. Now, it's pretty standard that I'll text once or twice just short notes about what's happening on things we've been working on. No big long discussions, just touching base. For me, it's a novel idea to be relying on someone vs always being the support for many family members so kinda a therapy goal. It feel felt weird at first, like therapy in general, but more natural over time. As in, I wouldn't ever be thinking to text my primary care doc or OB/Gyn, so learning to be comfortable with T took time for me.

I while I'm sure my T maintains similar contact with some of her other clients, I doubt it is universal with all.