In my experience, using the method that UTSG described will work, but…and this is just my opinion….will only result in a low sheen, satin blacking that doesn’t look like original Winchester high sheen blacking. Even Mark Lee recommends blasting the barrels or parts with a fine abrasive before applying solution. It’s hard to get a high sheen black when you have to do that.
In order to both polish the barrels to “chrome” and then subsequently get the slow rust solution to gently bite into the polished steel, you need to develop a method that allows that to happen. That’s a trick I’m not quite ready to part with quite yet.
I have not had good luck getting Mark Lee solution to bite into highly polished steel. It doesn’t work very well for me on anything polished to 320 or higher.

I purchased a solution recipe from a friend in Montana (he’s out of the blacking business) and I’ve seen Winchester model 21’s and model 12’s he’s blacked and they look amazing, correct sheen, etc. This solution has both Mecuric Chloride and nitric acid in it, albeit in very very small quantities. The solution gives a beautiful deep, high sheen black on highly polished steel. It usually takes about 10 to 12 cycles to get the correct color, on high chrome steel and nickel steel, a few more cycles to get there. He got the solution recipe and method from a long gone Montana gunsmith who also restored Winchesters. I personally have not used the solution yet on a set of barrels, only on test coupons of steel.

I’m thinking you’re stopping a little too early on your cycles. Have you tried going 10 to 15 cycles and seeing if the black holds longer? Also, maybe try letting the barrels cure in oil (like utsg said, non detergent type, I use non detergent 30wt compressor oil) a bit longer than 24 hours. I sometimes let the oiled barrels sit for a few days. Get a clean paper towel to wipe the barrels down and if you see dark stuff on it, let them sit longer.
Ensure you’re also using a saturated solution of baking soda and water to neutralize and let them sit in that for at least an hour or two. That was a great and very helpful tip from an old time barrel blacker that I got a while back. It makes a difference.