I don't see a long-term series happening (at least not for a bit) because it would sort of be covering ground similar to Strange New Worlds, but I certainly wouldn't mind seeing a miniseries.
I don't think a Stargazer series would come out anytime soon. I'm thinking
way in the future. Not only after
Picard is done, but also after
Strange New Worlds is done.
They keep saying they haven't rebooted Star Trek, so going to a Kirk Series after
Strange New Worlds would harshly contradict what they've told us because it would basically have to be a reboot (or else become the most hamstrung, mismatched series ever made). They won't do that. It goes directly against their company line and they don't need to do that if they have other options. Before someone chimes in to respond to this,
please read the rest of my post and hear me out.
When
Picard ends, people will still be wanting to see more of Picard even if Patrick Stewart presumably won't be able to play the role anymore by that point. And if you can't go forward anymore, you go back. Not including the horror flashbacks going on in Picard's mind in "The Battle" (TNG), we've never seen Picard's time on the Stargazer.
So you have a character there's more demand for, a ship already lined up, with a time-frame that's never been shown before, and a Quadrant that's not quite TOS but not quite TNG and the perfect middle-ground. The series is begging to be made. Eventually.