What about baking powder instead (or after) baking soda?
Much too soft to cut steel. Some 'abrasive' grits can function basically to knock loose bits of weakened steel (burrs, in other words) from an edge. If an edge is already very thin & refined, sometimes that's enough to produce a noticeable change in sharpness, in removing the fine burrs. But beyond that, they won't be able to actually cut the 'good' hard steel after the burrs are gone. This is true of 'polishes' like talc, baking soda/powder, most silica (including the 'hydrated silica' usually found in toothpastes), and sometimes compound known as 'jewelers rouge' (red iron oxide, a.k.a. 'ferric oxide'), because it won't be hard enough to actually cut most modern steels. They can still have a burnishing effect which 'shines' the surface a bit, but won't really be able to abrade, therefore thin, an edge.
For a so-called 'household' product which is able to really do some true polishing (of steel) by abrasion, look for metal polishes which contain aluminum oxide (such as Mother's Mag, or Flitz/Simichrome/Maas/etc). Most of the rest, like toothpaste, baking soda, talc (literally at the bottom of the Mohs scale of hardness) will likely be a disappointment, UNLESS the edge is already very refined, with just some 'cleaning up' of edge debris, like burrs or other junk left stuck on the edge by the sharpening process, making the only difference. If that's the case, then simply 'stropping' the edge on your jeans, or bare leather or paper or smooth wood, will likely accomplish the same thing.
A simple test to see if an 'improvised stropping compound' is actually sharpening (abrading) a knife edge or not, is to apply the compound to a piece of clean, white paper and strop on that (on a hard backing). If you can see dark streaks of metal being left on the paper, it's abrading the steel and may have the potential to actually sharpen it. If not, the 'compound' isn't likely helping or doing anything more than the bare paper could do on it's own; and that would be limited to scrubbing fine burrs off or otherwise just realigning them to stand straight.
David