Great answer, "Haters Make Me Famous". I personally agree 100%. But I would add, from a musical and creative perspective, that Hank Williams, Sr. was an incredibly gifted musical artist in every sense - in fact, an unparalleled genius. He expressed himself effortlessly and with precision (like the great John Lennon), and used musical rhythm (largely blues oriented) and chord structure to place nearly perfect melodies that reveal all the beauty, sadness, or frustration that certain chords and musical keys can evoke. In a way, every musical progression, key, and indeed every chord can have a "mood" to it. Hank brilliantly took these musical tools and created wonderful sonic environments with various themes, atmospheres, and feels - feels that capture very deep and universal themes of human experience. This feat is why he is deservedly hailed as the "Hillbilly Shakespeare".
Sometimes, it's upbeat, religious, and optimistic ("I Saw the Light"), or in celebration ("Jambalaya", as you mentioned), or in the shoes of another ethnicity ("Kawliga" and "The Funeral"), or romantically innocent ("Hey Good Looking", "Setting the Woods on Fire", and "Baby We're Really in Love"), or downright depressed and hopeless ("Alone and Forsaken", "Lost on the River", "I'm So Lonesome", "Long Gone Lonesome Blues", "I Don't Care If Tomorrow Never Comes", or the very mournful "I've Just Told Mama Goodbye"). Then there's also countless songs about romantic trouble ("Cold Cold Heart", "Your Cheatin' Heart", "You Win Again", "I Can't Help It", "There's a Tear in my Beer", "Take These Chains From My Heart", "I Hang My Head and Cry", "You're Gonna Change or I'm Gonna Leave", "Why Should We Try Anymore", "Why Don't You Love Me", and "I Just Don't Like This Kind of Living", among others).
At the end of the day, he was a musically advanced poet with a great sense of humanity in him. His songs are much more than what we usually tend to call "songs". They are little musical artworks - like any great painting or literary masterpiece. They tell us about ourselves. About Life. In them, we find much with which we can relate, and we realize human experiences are not singular but instead that they are what make us human to begin with. Hank displayed this humanity for all to see in his music (again, like the great John Lennon). He was a brilliant songwriter (and in my estimation, he is the greatest, even more so than Bob Dylan). Musically, creatively, poetically, and in every other way, he is a genius no one can touch, and the day he died, we lost a great wealth of possible musical artwork. We may never see another individual as uniquely gifted as Hank in a very long time. To me, he's the Micheal Jordan of not only country music, but of music in general. He was perfect. Excellent. High-functioning. Brimming with musical ideas and a need to express himself.
As a creative type myself, he's nothing short of a pure mentor and idol. A goal to aim at, knowing you'll never come close to reaching it, because only one human being has ever managed to climb so high (except perhaps Mozart, in my view).
But yes, you are correct: Hank does something to me that I can't really put my finger on. It's just there. And no one else does it. Just he. He was one of a kind and (in a sense) there'll never be another.