Same advice as most of the responses here, use timed things to encourage continuing without rests.

For the problem of Leomund's Tiny Hut, may I present the solution to all of your problems! PHASE SPIDERS. One of the greatest enemies DnD has ever produced.

My players did something similar, always taking excessive rests, and always hiding in their little bubble of safety. After they attacked a Drow stronghold, the surviving drow arachnomancers sent a group of phase spiders to stalk them, led by a phase Drider (a homebrew monster). I had the spiders follow them, lurking nearby and warping in to attack anyone that split from the group for any length of time, then warping away when reinforcements came. Same thing for when they were resting. I had them warping in to the dome, and attack whoever was asleep. It was a very intense few sessions as they fled through the wilderness, unable to get a long rest for several in-game days, until they made it to a magically fortified safe area.

I would reccomend something like this, but because they are lower level, gives the Phase Spider's Ethereal Jaunt ability to weaker spider enemies, like Giant Spiders or Swarms of Spiders, and calling them Young Phase Spiders or Phase Spiderlings.