TL;DR"Connect to Address" is quite simple and could remember more than only the lastest address used.
What ?My suggestion comes with two levels of implementation to improve the experience with multiplayer (re-)connecting to a private server:
1. Remember the password from the latest address and try it, if the address have not changed.
2. Extend the "Connect to Address" to a collection of private servers the player can manage. (Best: steam sync of this collection)
Why ?If a player is playing a lot on private servers, he has to remember or keep track of all addressess and passwords.
Reducing clicks and keystrokes to (re-)enter a private will bring more joy to those playing a lot on private headless factorio servers.