Just a thought or two:
I use Singer Sewing Machine Oil on all my high carbon steel edged blades and tools. I started using it after getting into fine Japanese swords and inspired pieces. The Japanese traditionally have used what they call Choji oil for centuries. This, turns out, is high quality mineral oil with a tiny bit of pure oil of cloves (for scent only). Singer oil is about the highest quality mineral oil on the market, not expensive, and readily available.
I picked up a couple bottles at JoAnne's years ago and still have one and a half bottles despite using this stuff all the time. I keep a large, flannel, gun bore cleaning patch lightly oiled with the stuff (meaning you only need a few drops around the patch, then fold and mush around to spread it) in my day and go-kits. After a few wipes on a separate cloth to clean off moisture or whatever, then a quick, light swipe or two with the oil patch and good to go. Great for Nihonto (and "Amerihonto"), but also my old carbon RMK's, FB's now, GB axes, etc. (a key is to not over oil the blade so junk won't stick to the oil).
Long, long ago (yeah like the '70's and '80's - sigh), for longer term storage or travel/use in really foul, wet weather, I used to coat the blades with a heavy gun oil like paste called RIG. Worked great for that. Not sure if it is still made. Might have a look, myself.
Also, lots of collectors and museums around the world use Renaissance Wax. Puts a great translucent fine coat of fine wax over the steel. Easy to score on the Net. Very good for long term storage and won't soak into leather or anything else so it stays on the steel.
Thanks, bikerector, for the Seal1 tip!!! Going to look into that, too.