Checking the staking wallet logs I do see that it continues to search its own outputs even if it accepts a block from a peer while searching. Here's an example:


In this case we didn't find a block, so the peer's block won. I'll see if I can find an example where JD found a block after hearing about one from a peer.

Edit: I can't find any. I suspect that although it continues looping through all the unspent outputs, accepting the peer's block has somehow made it such that no local stake will be accepted and so it's a waste of time to even continue searching in the current time slot.