I think 'jb' has the 'best' answer so far given everything you've said. Assuming the pipe insulation is in 'good condition', or better, and the temp is as you say, then I think the faucet position will tell you a lot.
As an aside, there are Water Saver Shower Heads. WaterPik makes them for sure. Be aware of a 'screen' inside the fixture for 'catching' any solids. Over time, you may need to remove the Head to clean it out. Read the instructions (if you buy one or 2), and it should tell you all about it. Using a WSSH may 'extend' the hot water for you the 2nd showerer.
My best to ya and hope this helps.
PS: After this afternoon, I won't be back until Monday, 9/18 but 'others' will be around ...