I wish there was some clarity on this issue. I'm checking out of a Club Carlson tomorrow in order to stay one night at a different Club Carlson property for the promotion. . . and then I'll check back in the day after tomorrow at my current hotel. I would rather just keep my existing room at my current hotel and double-up on rooms tomorrow night, but I'm checking out of my current hotel just to make sure I don't run afoul of some rule.
Incidentally, in searching for an answer to the question posed in this thread, I found that SPG does apparently allow multiple hotels for the same night. . .