Re: 05 taurus, Vulcan, starting problem with warm engine
If tune up is up to date (ie spark plugs, air filter, even an oil change etc) then I'd get a fuel pressure test done. Ford dealers (good ones at least) are good at diagnosing problems like this, as they see it often enough on their older vehicles.
It could be anything from a fouled plug or two, bad starter, crank sensor gone bad, to low fuel pressure.
It doesn't always work, but to try the fuel pressure test for yourself, before starting it after you turn the key, leave it in "run" for 15 seconds and then start it. This will allow the fuel pump to build pressure and if it starts without issue you've narrowed it down. A clogged fuel filter can also hamper flow enough to cause engine hesitation.
'11 Ram Quad Cab 4x4 SXT 5.7L HEMI (stock); 48,500m
'06 Pontiac G6 GT 3.5L 220hp; 64,000m
'12 Chrysler 200 Limited 3.6L 283hp; 28,000m
'99 Taurus 3.0L 2V Vulcan 145hp; 154,300m - Traded
Amsoil in all vehicles!