I case you missed it, the Republicans have devised a new stratagem to prove that all the bad things happening in the shutdown are really just the Democrat's fault. For example, the shutdown has caused the National Institutes of Health to close one of the experimental research hospitals which normally treats around 200 people including around 30 children with cancer who really have no hope other than experimental types of treatment because all the conventional methods have failed. Closing this cancer treatment program has been a bit of a PR nightmare for the Republicans because they are obviously 100 percent responsible for shutting down the government and thus closing this cancer treatment program along with a lot of other government programs.
The Republicans are now trying to pass a number of "patches" or bills to refund particular agencies and programs including the cancer program. The idea is both to make the GOP's shutdown look less bad and also try and force Democrats in Congress to take tough votes as reopening certain particular programs means that there is less overall pressure on the Republicans to end the shutdown. I doubt that re-opening a few small but highly visible programs will change the political calculations on Capitol Hill but you never know.