Car will start first time I drive it for the day, but when shut off gives issues

What is your question?

If I shut it off and try to start again within a short time period it takes a few times before it will go. And then a lot of times ( getting more and more frequent) when I stop at a light it kills and then won't start again sometimes for several minutes and to get it to start from that point I have to push on the gas a bit. We just changed the fuel pump, not the filter though, but we don't have a lot of money to put into this car. Can you please help me with some advice so we don't have to waste hundreds of dollars before the problem is actually fixed?

What makes this problem better or worse?

It seems to only happen after the car has been driven for a little while

How long has your 1999 Oldsmobile Intrigue had this problem?

Since we got the car about 6 months ago, we changed the fuel pump and that seemed to help for a while but now it’s doing it again