Can you recommend Jazz for some one that doesn't like Jazz?

Wow. If I were a guy who didn't know much about jazz and I asked that original question these responses, put together, would have left my head spinning!  Humor aside, if you like Mose Allison, you're pretty much already there.  

Maybe start by process of elimination, or simplify your selection process.  Given the wide range of musical styles referred to as JAZZ, start by identifying what appeals to you musically then choose artists that fit. For me, not really into big band, Dixieland, fusion, or experimental styles.  Jazz guitar, piano, saxophone, female vocals in smaller groups or ensembles are more my style.  Maybe call it more ‘traditional’ jazz.  Several artists listed in this thread fill my somewhat limited collection, meet my criteria, and I really enjoy during repeat sessions.  FWIW, several have passed but left prodigious musical  legacies.  The musical prowess and technical proficiency of the greats make most rock musicians sound like amateurs by comparison, IMHO. 

Yeah, numerous Mose Allison comments, he was a gem... moody, quirky sprinkled with humour throughout

The Mose Chronicles v2 in London* is not to be misses 

Yes I can understand you ,years ago ,I worked 11 pm to 8 am at the old PO.and the guy at the desk would play Free Form Jazz....I hated it...just blasting  note without any form of music ,Terrible....But you need other stuff... Older jazz Like  Chat Baker,Gerry Mulligan,The Rippingtons.Thats a good start, it's music,with form,classic songs....Enjoy my  Man and  have a Great Christmas....

@ezwind

Yes, the Dead’s jazz influences don’t show up as consistently as with the ABB, probably because their music encompasses a lot more different musical styles and genres. But you can certainly hear it in songs like Eyes of the World, Bird Song, Crazy Fingers, and some versions of Dark Star. And the band would swing a lot more when Kreutzmann was the sole drummer for a few years in the early 70s.

Sure. I don’t contest any of the above. Comparing the Allmans to Jerry and the boys, I meant, overall. I much prefer Kreutzmann on his own. IMHO, he didn’t need any help!  Like Mitch Mitchell, he has a "slippery" feel. I prefer this approach to heavy handed "bashers".