I am looking for a seat that appears close to stock that provides more support, yet makes it nice to have the top on for a 6'2" driver.

Or should I take out the rails, mounting it permanently?

Posted recently, take out the bottom seat pad and replace it with a thinner one. If that won't work for you.
By all means remove the rails and mount seat to the floor. You might want to make sure that the seat is correctly positioned for steering and shifting before you screw it down.
CCLINERGONOMICS101

I saw that, thanks.

Are there any aftermarket seats that are close to stock in appearance but provide more support, comfort without loosing leg room or head room.

Price I am less sensitive too

Modifying the stock seat is nearly always the best approach. There are relatively simple ways to modify the stock seat pan to add side support, and just removing the bottom cushion makes a big difference in height. If you look closely at the seat and mount, you'll see that sitting on the bare stock shell is the lowest it's physically possible to go and still keep fore/aft adjustment. If you lose the sliders and rocker, you can go a bit lower still mounting the seat directly to the floor and crossmember.

Most of the aftermarket seats actually have a thicker than stock bottom squab, so they position you higher, rather than lower. Most of them also have soft webbing below the cushion, and a pair of steel bars or straps connecting the sides together. Removing the cushion on one of these positions you in the same place as removing the cushion in the stock seat.