I typically put some Hoppe's 9 on the bristle end or just before the bristles and after a few passes would put some oil on the tail end.
Ideal would be run a patch soaked with solvent and let it sit, then use the snake, but if you are using patches with a rod/cable it takes away from the main benefit of the snake, which is convenience.
As others posted above, I got disillusioned after passing a patch (with or without solvent/oil) after boresnaking and was surprised how dirty it came out.
They are good for quick cleanings at the range.
They will get the bore as clean or cleaner than you NEED it to be, but if you really like to get your bore clean regular patches/brushes with a rod or cable (like Otis) I think works better.
ETA: for OP, what you are doing sounds perfectly fine. I normally add a little oil at the end. On chrome lines barrels there should be no need to do so, but it is a habit I got into where I like all metal in my guns to have a light coat of oil.