TinyCore is just Core with some extensions included, the repo is the same. Also note that you don't have to install the same system you installed grub with, you can create your usb stick with a live corepure64 edition and have it boot 32-bit core.
Some history with this, grub 0.97 could not be compiled 64-bit, so if your system was pure 64-bit, you needed to use a 32-bit system to install grub.