I'm very sure sure it's a freewheel, not a cassette. If it's a freewheel, new 6 and 7-speed freewheels are still readily available and a 7-speed will fit in your current dropout spacing. Further assuming friction shifting, the 7-speed will work with your shifters.
Here is one source for 6 and 7-speed freewheels but there are a lot of other sellers as a Google search will show:
https://www.sheldonbrown.com/harris/freewheels.html
If you do change the freewheel I recommend a new chain to go with it.