I occasionally shoot with a gentleman who uses a cane to walk. He has a leather strap(string would work) with one of those spring loaded climbing type clips through the leather strap. He then clips it to his belt loop while shooting. This way he doesn't lose track of it or drop it.
Sounds like you have one of those new "T" handle alum. canes . Get rid of it and get a old style wood cane with a curved handle on it . Use the curved handle to hang it in your hip pocket . Easy to reach when you have to move but out of the way when you have to shoot . Been working for me for over 40 years .