I used to keep an oily rag for giving a wipe over of tools such as pruning saw, secateurs, axe, shears etc after use but it was messy on the hands. I recently had an appliance delivered that had a couple of small bits of sponge plastic as packing, about the size of a very large bar of soap. I put a few drops of ordinary car lube oil on each and keep one in the greenhouse and one by my tools. Excellent for a quick wipe over. They are reasonably rigid, so its quick, and little or no oil on the hands. One of my unexpectedly better inventions - for free.
I have similar oily rags, but keep them in plastic bags (freezer bags are good). When I use them I only hold the outside of the bag (pulled over my hand), and keep my hands clean ...
I use stainless steel tools, much easier