What are the strategic benefits of a ship teleporting out once they engage an Opp ship? Buying desired things in one Sector, and then zipping out to fight an Opp ship in another Sector, obviously. In general, though, what else...?
The engage phase happens all at once simultaneously, so no, once everyone picks an engage target, there is no repicking later. There is also no card plays allowed during the engage phase, you must do it before or after the entire phase (similar to how every sub-phase works in the game.
The benefits are rare of engaging a ship and then teleporting out before the damage calculation phase, mostly because you don't get to deal it damage.
But maybe, for example, The Flagship is alone in one sector and none of your team can safely take the damage from a fight, but you also dont want a specific card from that sector to flip in the aftermath. So in the engage phase you target the flagship, then before damage calc you fly out so as to prevent that specific flip. So there are some edge cases where this tactic could be useful
You buy things before the engage phase, so not sure if purchasing stuff in a sector plays into the decision of leaving after engaging, but maybe I don't quite understand your question correctly.
What I meant was that there's a Tech you really want in Sector A, and youi need it to fight an Opp ship in Sector B. So you go to Sector A, then blip out and fight the Opp Ship in B later.
I mean sure you can do that. Claw and Marrot are both strong at this since they have ways to move around between phases in their decks. But if you get the the engagment phase and still in Sector A, you must engage a ship there. After the phase ends and you're allowed to play Boosts again, you can travel to Sector B, but you cant re-engage another taget since you used your one engage. That make sense? Of course you could always move out of Sector A inbetween the Requisition and the Engagment phases, but then you wouldnt keep an opp in Sector A engaged and free it up to flip stuff in the Aftermath.