I've been thinking over your problem and in my opinion more then likely the fuel filter is dry or dirty or clogged and this is why you leaving the key in the run position slightly helps bec the pump is forcing gas through the filter , also from what you're saying if it sits awhile like an hr or two the filter starts to dry up again and thus starting issues one again.
The chances are that the fuel filter hasn't been changed in years on that vehicle bec on the average few ppl change the fuel filter once a year most don't change them or very rarely change them , plus any used vehicle should have it changed asap as well as having a basic tune up and treating the fuel with STP or Lucas etc gas treatment.I'd try another starter relay in it just in case but I doubt that it's the relay.
In order to have any engine turn over via the starter fast and quick enough to start the engine up right away the charging system from the battery , starter, alternator , engine block ground posts and all the grounds should be clear or any rust or corrosion.I'd go over these connections one by one including the o-ring wire connects use a file , a wire brush , sandpaper and clean these connections fully and then use automotive anti-moister contact protector jell on them.I bet that unless someone kept real good care maintenance wise of that vehicle that these connections are indeed rusty or corroded and could be adding to your starting issues etc.
Personally I find it odd that if a shop changed the fuel pump and a hose etc and they knew it's a vehicle that age and just purchased used that the mechanic wouldn't automatically recommend and change the fuel filter bec basically whenever that type of work is done the gas filter should be changed asap.But again this is only my practise and opinion but most good mechanics do change the fuel filter when performing gas tank or fuel pump or fuel line etc type of work.
I'd also have the battery, alternator charging system checked many shops will offer free charging system tests just be sure that if any problems do exist that the mechanic show you via the test equipment etc and fully explain the problem.Example don't book the charging system test and tell them that you've starting problems just say you want a full charging system test to be sure that all is proper etc.Then see what the mechanic or shop ends up telling you and compare it to what you know is occurring.
I extremely recommend that anybody that buys any vehicle and wants to keep it up and learn or repair the vehicle to buy a service manual even if it's a walmart you buy it from and pay 20 bucks or so for bec they are a great tool not just to help and guide you but to teach someone much more hands on about their vehicle or at least it helps them understand much more about their vehicle, plus owning a 12 volt tester and a voltage/ohms meter is also a great idea.
Next I'd consider the vehicles age and regardless how old it is unless it's brand new I'd be sure to go over it the belts the fluids , changing the vital fluids so you know that all the fluids like the oil or coolant or transmission fluid etc is clean and at the proper levels and of good quality.Many extra parts can be purchased via a u pick auto wreckers etc and you'd be surprised in how many good used parts are out there.
Example on that vehicle you've spark- plug wires , injectors , ignition coil or module and pvc valves and sensors, air filters, belts , bulbs, fuses and relays etc that all could be bought if you took the parts off yourself or with a friend etc for quite cheap.If you're lucky you can even find whole almost new spark-plug wires and modules etc and its great having a exact matching set of plug wires bec then you can use a ohms meter and compare readings to the wires on the vehicle and the extra set you have when you're performing a tune up etc.
Hope that helps and best of luck.By the way one reason my answer is so detailed is bec I've owned a grand am before and once a did a full tune up , balanced and changed the tires, wheel alignment etc that vehicle drove like a real smooth ride and was very good on gas , plus an excellent highway driving or cruising vehicle. Really besides my old monte carlo my favorite car was that grand am , also I've owned many GM's and other makes of vehicles , plus I've fixed many vehicles over the years and driven some cars several hundreds of thousands of miles.
There's nothing quite like knowing your vehicle and doing the work mostly yourself and my best advice is watch your dash gauges etc and get to learn just how your vehicle over all responds , gauges and all when cool or warmed up and when driven for many hrs at end etc.
