But anti-tax sentiment has turned out to be a patchwork in this county, which is about the size of Connecticut, with just over 100,000 residents. In recent months, some communities voted to pay to reopen or support a town library, while others insisted that volunteers alone would suffice. The result has been more tumult: A split between rural parts of the county, which mostly rejected higher taxes, and urban parts; an us-versus-them battle over who now gets to borrow library books; and general chaos, as people try to figure out the mechanics of running an institution that had long been the purview of local government.

Douglas County, deeply Republican in a Democrat-leaning state, has the fourth lowest property taxes in Oregon, according to state figures, and a county library tax would have added about $6 a month for someone with a median priced home. There are also pockets of rural poverty in towns like Glendale, population 800, where a branch library was kept afloat in the old days by a countywide sharing of resources. But ever since voters rejected the library tax and the libraries were shuttered in early 2017, homemade efforts to reopen them have cropped up.

“It’s keeping me awake nights,” said Betsie Aman, a substitute teacher and volunteer at the library in Glendale, which reopened for 12 hours a week as a nonprofit corporation with an all-volunteer staff. Among a core group of women who led the effort, three have withdrawn because of illness, advancing age or fatigue. “We’re getting kind of burned out,” Ms. Aman said.

Legal and logistical issues have made the struggle harder. Douglas County retained everything in the stacks, from books to videos, making it difficult for local groups to take legal possession. Some of the small groups are hashing through questions they never needed to think about before: grant applications, training, even rules about family leave and retirement.