They ALL get hot with extended shooting sessions in 308. Assuming your handle means you have a ruger scout, it is just as good might better than most any remmy barrel being hammer forged. Go to remingtons site and scroll around till you find what you want. When you can not set your hand on the barrel and leave it thee it getting hot. heavy barrels hold up better than skinny but they to get hot just take a fw more rounds. If you want a high round count per session buy a AR in 5.56 with a chrome bore or a melonite treated bore.
Have your gunsmith put a MTU (ie. #7 Bartlien, #10 Kreiger, etc...) barrel on it. While he is at it have him true the action and do a bedding job on it. Badger Ordance makes a good DBM bottom metal which you can get 10 rounds mags for and McMillan, Bell Carlson, Manners, etc... make a good selection of stocks.
I recommend buying an older used Remington 700 with a bull barrel that was manufactured before the Freedom Group bought Big Green. Remington's quality control has slipped of late (I had to send two Rem rifles back to the factory last year). The bull barrel will get hot, but your groups will not open up as they would with a sporter barrel. On the plus side, my 1997 Rem 700 ADL in 30-06 never failed to function, handled like a dream and could place three rounds in a hair over an inch (so a good marksman could have shot sub MOA with it).