I've got that Tagarden/Satchmo duet on CD ~ it is really great.

I have the impression that none of the "Rocking Chair" songs we've mentioned are the one being sought by Lynn ~ not the old standard, not the Band song, not the Rick Cardwell song, and not whatever is sung by Jerry Jeff Walker. There most be more...

I just Googled ["Old Rocking Chair" = lyrics] in an attempt to find the composer of the "standard" ~ I think it may have been Hoagy Carmichael, but don't know for sure. I found it listed under the name Ted Heath, but Ted may have been the bandleader/recording-artist of one rendition rather than the writer ~ most internet lyrics sites ignore the songwriter credits and credit songs to performers, and this source could have been one such site.

I also learned that there's a C&W song "Rocking Alone in my Old Rocking Chair" which was recorded by Eddy Arnold, Hank Snow, and the Everly Brothers, plus at least one other song featuring the phrase "old rocking chair" which is entitled "Home Among the Gumtrees." ("Gumtrees" mentions Vegemite, so I'd guess it's Australian.)