There have been a few times when I needed a bit more accuracy, such as at the unmarked intersection of two wickedly twisty roads in the middle of a Virginia forest. Even with a good compass of course you're left with a bit of guessing.
The only time I can remember something like that happening, I was more confused by the resolution of the map than by the compass. I knew darned well I wanted to go 45 degrees true, and the compass was good enough to show me where that was, but my choices were 0 and 90 degrees!
I have just carried a hand held compass. It is a very basic Silva model. The thing is that in the touring I have done, I don't recall using it even once. I have found that I have only needed a very general sense of which way is north and that can usually be maintained even on cloudy days.
That said even if you actually want/need to use a compass a handheld one might do just as well. The few times where finer accuracy is required stopping and digging out a handheld should not be a big problem.
If you need a constant reminder of direction as you ride a compass with marginal accuracy ought to suffice. Personally, I typically find that the direction of the light, wind direction, and various land marks (mountains, rivers, lakes, oceans, cities, and so on) are sufficient to keep me generally oriented.
For off road touring I would probably need a compass more, but a handheld would still be my choice.