There are so many flights to Phuket everyday, if you land in the morning then you have a great chance. But you might have to wait all day or upgrade to buisness class (about 1000Bt more) Check it out on
Thai Airways
Thai air have themost frequent and reliable service. the cheaper airlines are often significatly delayed and not infrequently cancelled. Thai very very rarley cancel a flight on this route practically never. You can call your local Thai office and make a booking and even pay with your credit card. You get an E ticket. Very flexible.
I would check availablity 1 month before on line then 1 week before and book then if it looked busy. if you wait until just a day or two before you can book and then pay for your ticket at bangkok. Or if you go through a travel agent then they can make abooking for you and you can pay just before you travel.