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Basically either before or during the upgrade process, the old image will need to be removed if there's insufficient flash space for both old and new IOS images. Of course, this does create a risk if upgrade isn't completely successfully in that a reload/reboot will not have a valid IOS image.
Some of the later IOS image just won't fit on the earlier 3560/3750s due to insufficient flash space.
You can (somewhat) reduce the flash space requirement by using the IOS .bin image rather than the full IOS .tar image which also includes WEB management. If you use the .bin image, you need to manually do much more setting up the IOS than just using the archive command.
This reference provides information for upgrading IOS on a 3750, http://www.cisco.com/en/US/products/hw/switches/ps5023/products_configuration_example09186a00804799d7.shtml, the 3560 is about identical except it doesn't stack.
