if your console is softmodded, you should be able to cold-boot your xbox with the game you wanna play inside (it must be an original, no backups) that way it will boot your game's default.xbe and will continue running the original non-modified bios. If you want to use your softmod again u just eject the game or start the xbox without a game inside.
Here's a scheme of the booting process:
if default.xbe found on D:\ run it (bios dont get patched so u have clean access to XBL), if not run xboxdash.xbe which runs the hacked fonts, which run the bios loader or kernel patcher which run your homebrew dashboard (eg. UnleashX, EvoX, AvA, etc)
Of coarse, this could be a bit different for different exploits, but ANY softmoded xbox will run D:\default.xbe without patching the kernel or replacing BIOS, allowing u to connect to XBL