Fuel problem -- 99 Elantra

Hello,
    On my sister's 99 Elantra (112k miles) she's having a problem starting the car after the car is warmed up. In the morning she starts
it with no problem. After the car has been driven for a while and then is shut off if you try to start it again the engine cranks and eventually will start but begin to idle roughly. When you give it enough gas, the issue goes away. I'm wondering if it was a clogged fuel filter, but if it was wouldn't it do this problem each and every time? Could it be a failing fuel pump? Any help would be appreciated.
Thanks, Nick
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My 2 cents worth is that it's most likely a sensor that malfunctions when warmed up. Does it set any codes? Possible also to have a manifold leak that is more pronounced when hot. Did you check the TSB's on HMA service? BCinBC

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Unfortunately no codes are being generated. I would have hoped there were as it would make life a little easier.
I've checked the TSBs on fuel and rough idle but didn't see anything that looked like it was this problem.
wrote:

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It's doubtful the fuel pump is the issue. Fuel pump issues are normally most noticeable when the accelerator is depressed, especially when driving.
The coolant temperature sensor comes to mind as a possible cause. But we're still talking about low-probability guesses. This may be best left to a professional at a good driveability shop with a good scan tool, at least for diagnosis.
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Thanks for the information. I was reading on HMA Service about it. Wouldn't it cause the check engine light to come on if it was defective?
On Mon, 12 Mar 2007 16:53:04 -0500, "hyundaitech"

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In many cases, yes, it'll trigger the lamp. But if it's just a little off, maybe not.
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In most cases, the sensor will indeed set a check engine lamp. Typically the failure mode is an open circuit, which the computer easily detects.
If, on the other hand, the sensor is just off 10 to 20 degrees or so, it's much more difficult for the ECM to pick that up.
I don't thing the coolant temperature sensor is a high probability item. It's just one of the things that should be considered.
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