Okay, so to confirm- attempting a grid hop (without permission) starts the timer, but does not increment your score. Yes?
edit: as a corollary, you could still attempt the grid hop many times with minimal. Since it doesn't add score you could easily do like 20 attempts in like a minute. Obviously that cuts down on your remaining time to do other things, but compared to the normal consequences of illegal Matrix actions grid hopping is far less dangerous.
2nd edit: Since you can reboot whenever unless you're link-locked, if your first illegal action was to grid hop, you could phrase your instructions to the GM as such:
"I attempt to grid-hop. If I fail, I reboot. Then I try again. Repeat until I hop grids on my first try."
That would conceivably always get you into the grid on your first try, maximizing your remaining time until the first increment.