How do I prevent the outline from dipping into an image that has gaps inside it?

[denzjos]

I think you used the fuzzy select tool to isolate the snowman. If the color of the gaps is the same color of the surroundings (can't see because the original is not posted) it is a little tricky with this tool. If the color of the gaps is a little different you can deselect everything under the tool options ( antiliasing, feather edges, etc...if you first select the fuzzy tool) and try different values for the threshold. If that is not the solution, you can paint the gaps with the paintbrush tool, not much work, it is a simple drawing. You can always use the scissors select tool, but that is more time comsuming.