When a player holds more than one instrument, how do I get the names in the upper left of the parts to stack vertically?
You may have to create a custom First Page Template and replace the part-name token with a text field.
As the layout name token shows exactly that: the layout name, you can go in Setup mode, and edit the layout name on the right panel inserting line feeds at the desired place (you could for example write the Layout name into a text editor and brake the lines with “return”, then copy and paste the text into the layout name.
You may need to make a customised First Page Template for those layout/s, but just to accommodate the higher text that will be generated.
See this post as reference:
One way is edit the Page Template and replace {@layoutName@} with {@instrumentList@}.