Uh, what? Your character logs into the location that you were at currently - unless you are in a "My Duty" menu activity. Unless something has changed since I logged out last, I have characters parked in my base, in the HQs, and at their last locations in the open world. If your characters are showing up somewhere else, I would suggest changing your password ... and use the log out option on the launcher if the problem continues. It's more likely that someone has access to your account than the game is moving your characters around when you are playing them. If you don't log out your account in the launcher, anyone else that uses your computer has access to play the game with your characters.
you can actually simply be in that area BEFORE logging out and ta-da, once you log back in you will be there upon logging in again.
Masks, you missed the point of the OP. He is wanting an option to re-log at a specific place (ie: He logs out in the middle of Gotham but when he logs back in he would log back in at his Base). An odd request to say the least, but that is what he was asking about.
Why ks it odd? Lets say you get linkdead in a zone infested of enemies. Would you want to relog there or in the safety of your base?
It won't happen. That would bypass the warp base / HQ cool downs. You would be able to log out and be in WT without addressing the warp cool down. Sounds like a way to get around a teleporter