Maybe Hank replaced the original Fender strings on 34346 when they wore out or broke, then had to put the paper in the nut slots to stop rattling British replacement strings, as seen in a few photos?
I would guess that many types of strings were always available in America back then - Gibson, Black Diamond and Fender at least all came as roundwounds as well as flatwounds.
I would think nearly all new US electric guitars came with flatwounds from early on, as they were considered to be THE professional high tech (hi-fi)solution to string noise. Watching and listening to the old US TV shows like Town Hall Party, you can see and hear the top players like Joe Maphis, Larry Collins and others using flatwounds. There's something about the way TV/stage lights reflect on smooth flatwounds..
Not all guessing on my part either, for when I landed in Australia in 1963, there were always American strings available, in flat and round, often new old stock.
I can remember buying flat and wound Black Diamonds for instance. The UK light gauge roundwounds came out here a few years later, but shortly before the USA brands.