Baseball question

This is a really ticky-tack question, but it bugged me the whole time I was mowing lawn today. Based on what I know from watching MLB on TV, starting pitchers who have the day off sit in the dugout, while relievers sit in the bullpen. Now, pitchers who are normally starters get called to relieve in rare circumstances, such as game 7 of the Series. My question is where those guys sit during the game. If they're in the dugout, they'd have to run across the field to get to the bullpen to warm up. But they wouldn't already be in the bullpen, unless the manager anticipated them being used, right? Is there a secret tunnel connecting the bullpen with the dugout/locker room?