I think I remember one of the talking heads on the Ken Burns Jazz documentary crediting Louis Armstrong of putting the blues in Jazz so it's been a part of it for a long time. I thought relating jazz to something you know could demystify it a little. I'm a novice player and my little discoveries are part of the joy in life, so to the extent I can pass something along I'm happy.
Examining Tommy's bass line, I hear a standard blues I-IV-V chord progression built primarily on the 1-3-5-6-5-3-1 pattern you describe, played with a shuffle feel. A great Texas tradition. I don't know if there is a similar "standard" progression in the jazz blues tradition, but the above 12 bar progression for Paul Chambers' blues in Bb as interpreted by Jim Stinnett goes:
I-IV-I-I-IV-IV-I-VI-II-V with a two bar I-VI-II-V back to the I. Unlike Texas Flood, the jazz blues line is of course played with straight quarter notes, four to the bar. That turn around is a signature of the style. More chords, delayed resolution back to the one means more interesting tension and release (to my ear), maybe more opportunity for interesting musical possibilities.
Stinnett doesn't mention where that line comes from, but I've played it along with Sonny Rollins Tenor Madness, and it fits. (I use the YouTube tools to slow it down to 3/4). It's a blast playing with Rollins, Coltrane, Garland, and Jones. Like taking a joy ride in a borrowed Lamborghini. I urge you to try it! It's just 12 bars, and I play the same line over and over to every solo, but that's o.k. It's actually cool listening to how different the same line sounds with the various soloists harmonizing over the top. I don't know of a more fun and painless way to see if this kind of thing is for you.