From tabletop roleplaying games, to simple random draw needs, I've always felt the specific requirement of a higher than six-faced die a bit limiting, especially when the easiest dice sets to find around are often pairs of regular six-faced dice.
So, how can we make use of every possible combination of a two six-faced dice roll and get bigger-than-12 results?

Here's the most viable solution I came up with:
to obtain up to 36 unique results from a two six-faced dice roll, one of the dice keeps its faces as is, while the second die has its faces replaced by new values (through stickers and/or markers).

This is the generic rule to follow when assigning the "expansion values" to each face (no matter the face-configuration chirality of the die):

These positions preserve a constant top-bottom sum of 30, just like the constant top-bottom sum of 7 on a regular six-faced die.
After a roll of both dice, the sum of the value on the expansion die's top face and the value of the regular one's will always provide a result between 1 and 36.
I've personally tried it with square pieces of white masking tape and the use of a permanent marker to write the expansion values.
If well cut and neatly applied, the duct tape's effect on the roll's fairness is negligible.

In any case, this kind of solution is probably the only one able to overcome "emergency" situations where a +6 polyhedral die isn't readily available.

This is what the pair of dice looked like in my test:

So, to recap, the main advantages of this system are: