Since log file access is within the logging program, you can work multiple contests at the same time. You load each contest by just clicking the filename. There is no need to re-start the logger. When I`m logging in a US contest and hear a DX station I want to work, I`ll click the HF log file and work the QSO. Then click the contest log file and go back to contesting. The LOG filenames must be different.
Of course, you need to start the logs that you want to use - select the contest, give it a filename and <Enter> at least one QSO before starting or loading another file. Moving the older log files to another folder will make this easier. Using descriptive log filenames is helpful - 2012NJQP.log 2012MARAC.log for examples. Submit both logs to the contest sponsors.
A multi-contest effort requires some thought post contest. Although the actual LOG files are different, at the end of the contests, the post contest files that you write will overwrite each other - urCall.log, urCall.sum etc. So, e-mail the file(s) to the sponsor one contest at a time.