Of course, if there's a black hole on the extended runway centerline, I wouldn't go there!
Seriously, you make the best choice available.
So you're following the SID, what altitude are you going to climb to? Say for instance a SID from Heathrow, you're going up to 6000 ft. Is that really what you want to do if you had an engine failure? I mean, we're talking engine failure/fire here which is something you'd return to the airport for, right?
Do you brief what altitude you want to climb to (usually I brief the nearest round altitude above applicable MSA) beforehand? Otherwise you'll need a second call to ATC to tell 'em what altitude you'd like to stay at.
BUT, you're following the SID and either climbing up to 6000' (not advisable if you don't know if the fire's out and the engine is shut down) or staying at a lower altitude, and now you might bust certain altitude/waypoint restriction on the published SID.
All things to consider.