An idea:

Flag check boxes for all six places (below 1st). Players could check any box below their current level.

This would remove the issue with checking a box for, say, 2nd, and then having the box change to 3rd place exactly when you check it thus giving you a 3rd place flag.

The other nice thing is that players could leave the game immediately by flagging last place when they are technically in 3rd place (for example) but they know they are really in last place because of scattered territories, low dice stacks, etc...