These are all the hut routes I know of for the huts listed on huts.org (including 10th Mountain, Alfred Braun, Summit, Grand, and private huts). If you have any additions or corrections, please feel free to reach out to me at jeff.j.suda@gmail.com.
The routes are color-coded as follows:
Red: Standard route
Green: Hut-to-hut route
Black: Alternate route
Please note that the Section House to Francie’s route and the Sangree M. Froelicher To Carner's Cabin are potential optios, but I couldn’t find any evidence that it has been completed yet. Similarly, the Sister to Section House route appears viable based on satellite images, but I haven't confirmed it in the field.