I would get a stick rule with a depth gage to begin with, because I think every woodworker would benefit from having one anyway. Using the depth gage, you can open it up to the longest length that fits inside of the area you are measuring, and then extend the depth gage the rest of the way.
A very low tech way to do it is to make some measurement blocks in say 1" increments, then you can use that plus your tape measure, or stick rule to make inside measurements, that way you can measure to the edge of the black instead of having to bend you tape and guess.
I think beyond that, a decent set of calipers is a great investment for making accurate thickness, and depth measurements, and can make very accurate inside measurements up to its limits, depending on the size you get.
I like to remind myself sometimes that you can build without a tape measure at all with the right techniques.