With all but the simplest of steels (1095, CV, 420HC), I've usually been disappointed with stropping attempts using materials not created for abrading hardened steel. The abrasive in Brasso is silica-based, and it's effect on hardened steel is negligible. I've NEVER seen toothpaste do any good on any knife steel (won't remove any metal at all; a bare leather strop works just as well). Barring the availability of AlOx compounds, SiC grit or diamond paste, I usually rely on touching up only on a diamond pocket hone (for 'field' use), and then just clean up any burrs on the inside face of my leather belt, which has some green compound applied. I've gotten to a place with my honing skills, I seldom feel the need for a strop if it's not actually within arm's reach. I can get by without it, if need be. I do have a favorite denim strop at home, and use it with some white rouge; it can do nice things for an edge, when it's conveniently at hand.
I did get lucky once, in finding a nice piece of natural sand/siltstone while on a walk one day. Very fine grit; the stone worked very well as a waterstone, in rebevelling a 420HC blade (Case) and leaving a nice toothy edge on it. I also used some of the 'mud' from the stone on a cheap bare leather strop, to clean up the burrs. That was a very satisfying experience, in finding something literally picked from the ground that functioned well in the full task of sharpening a blade to full edge restoration.
David