So here is my suggestion: Add fleet commands to combat phase and a real fast forward mode.
1. Fleet commands
a) Attack >> Will cause the fleet to engage the closest enemy so I don't have to move it manually. This is number one choice in a fleet on fleet battle with only 1 fleet on each side.
b) Move to position >> This will move the whole fleet at once to a desired position so I don't have to click every single ship. This is a good choice for tactical combats where I have multiple fleets attacking multiple targets.
c) Avoid combat >> This will cause the fleet to move on the edge of the sector trying to stay out of battle. That's my choice for non combat ships like troop transports and colony ships when I want them to stay in sector.
d) Retreat >> Will cause the fleet to flee from battle at the shortest way possible (meaning to leave the sector). This choice is for non combat ships that find themselves facing warships accidentially which occasionally happens to my unarmed scouts.
Ship activation behavior for active fleet commands
For commands a) and b) only the first ship of a fleet will be activated every turn until the command is executed to give player the chance to change the command.
For commands c) and d) the ships are only activated if one of those ships is enganged by an enemy ship. This will not cancel the command automatically so player can decide that his ships are fast enough to escape. Unlucky crews if they aren't

UI for fleet commands
In the UI for a selected ship there is a tab control for the ship commands. I suggest using the (currently unused) second tab to switch to fleet commands and moving the "fleet focus" button at the bottom of the first tab to that second tab. This will give five buttons on the second tab which should fit really fine.
2. Fast forward mode
Currently auto combat is pretty limited. The only way to achieve my goals in combat face is to do everything manually - which is far too much time consuming especially in the late game phase where I have lot's of combats. So I want to have a fast forward mode that respects player's choices like the auto turn option on the galaxy map. This means the turns are switched automatically with the player's choices until something changes, for example a ship getting in weapons range of or being attacked by an enemy.
UI for fast forward
I suggest using the existing skip to end button for activating fast forward and changing the auto mode button behaviour to the current skip end button behaviour. You could add a checkbox next to the auto mode button for enabling it for future combats to get the current behaviour again. This feels more naturally for me as I lost more than one fleet because I wasn't fast enough in realizing that auto mode is still active from previous combat.