I'm still a little confused, though. I looked at the map you kindly linked for me, and it has a red circle about 40km NE of Tarnow, but I still see no mention of Tarnowce. Are you saying Tarnowce is a region and these other places you mention are part of it?
I'm assuming Słupiec, gm. £ubnice, pow. Staszów, woj. świętokrzyskie are places - I'm a only a week into teaching myself Polish and so still mumbling Nie, Tak, and Dzień Dobry at the moment. I get funny looks on the bus :-). So, sorry if I appear a bit simple with that.
If you, or anyone else, could expand on it I'd be grateful.
You've got Tarnowice in the heading and Tarnowce in the text. Couldn't find Tarnowice, Tarnowce is some tiny obscure village, but there are six localities called Tarnowiec. Could that have been what you were looking for?
Incidentally the "tarn-" root comes from tarnina (sloeberry) -- the stuff Brits use to make sloe gin.
I noticed that on the map from Peter's original link. I did wonder if that was what io was looking for.
But now I'm pretty sure I've found it very close to Przemyśl, right by the Ukraine border. The Ukraine element fits in with other information I have about the person I'm tracking.