If there is oil residue in the barrel, the first shot will probably be a flyer. To minimize this tendency, wipe the barrel out with bore cleaner before you go.
Some, not all, barrels will give tighter groups when the barrel is a little fouled or warm.
As for letting the barrel cool; it depends. For a very over bore cartridge like the 7mm STW, 220 Swift, etc. with heavy loads, if you want to maximize barrel life you should probably let the barrel cool between strings. For more common cartridges like 30/06, 44 Rem. Mag., etc. it doesn't matter that much.