I did remove the rubber piece and old adhesive, and re-attach it with new adhesive, but the procedure was more difficult than anticipated.
First off, it was difficult to scrape off the thick layer of the old material, once I had easily detached the rubber piece. Isopropyl alcohol, lacquer thinner, etc. woul not budge it. Finally, multiple applications of citrus-based Goo Gone plus scraping with a knife did the trick. Evostik is not available in the States, so I tried DAP Wedgewood contact cement, which would not stick to the rubber. Two of my old standbys: Household GOOP and double-sided carpet tape (you peel off both sides and are left with a sticky sheet) did adhere adequately, but within a hour each had become unstuck either to the rubber or to the metal base. Finally, Scotch heavy duty mounting tape has held for 24 hours.
Hope this helps other DIY'ers.
Ed