Runs fine once it's running

Sounds like a bad coolant temp sensor very common on those cars. Do you have any codes?Have you checked yet?Next time it wont start hold the gas pedal all the way to floor while cranking see if it starts.

You dont pump the gas you hold the gas pedal all the way to floor while trying to start it. Also just because the service engine soon light isnt on doesnt mean it doesnt have codes. Check for codes also have you checked the coolant temp sensor yet?Thats most likely your problem.

What color was the connector end of the new sensor where you hook up the connector?Also did you inspect the terminals of the connector for being corroded?Try this when it sits over night before starting it try turning the key to the run position let it sit there like 5 seconds without trying to start it do that two more times then see if it starts. You have a 1997 saturn S model car those fuel pumps are known for not holding the fuel pressure in the line and bleeding down. Causing hard starts like you have. You can also get a fuel pressure gauage and we cancheck it that way also.

The white connector ones are usually for the temp gauge on the older saturns and dont work like they should as coolant temp sensor. Also auto zone rents the fuel pressure gauges for free with a refundable deposite.

Sounds like a fuel pump bleeding down you will habe to rent a fuel pressure gauge to prove that. Also what is the exact code number your getting not the code description.