This is just off the cuff, but what if whenever you missed a HBTD roll, you added 1 to future rolls until you made the roll and it then reset?
For example suppose you start out with an 8+ to make and you roll a 7. So you miss the roll and suffer the consequences. Then the next roll you roll a 6. You add 1 to the roll for 7 but you still miss. Now you have +2 to your next roll. Then you roll a 6, add 2 and get an 8 so you make the roll. So now you're back to +0.
That means if you have a bad run of luck, eventually the pluses will be so high that you will make the roll, at least once in a while.
The effect is to slow the timer to a very slight degree, but the greatest effect is only when the players are having repeated bad luck.
I'd prefer using something like that rather than the blind die roll we have now. Taken a step further, it would've been awesome if the HBtD roll was modified by how well you're doing in the mine so far:
-Pass your last Encounter? Gain a +HBtD Token, making the next HBtD easier. Fail your last Encounter? Gain a -HBtD Token instead.
-Found 2 Clues so far? Gain 2 +HBtD Tokens.
-Killed the Miniboss already (would have to actually add balanced miniboss encounters to the game first)? Gain 2 +HBtD Tokens.
It would be a completely different system, but something like this would give the feeling that you're going to succeed or fail your mission based mostly on how effective you've been at delving into that mine, rather than how unlucky you've been rolling a 2d6.