This is not my list but I've eaten @ most of them. None of these are within walking distance of downtown hotels but several (Sam's, Smash'd, & Burgers by the Butcher) are near the Pearl if you find yourself in that area. Chris Madrid's is probably the best known outside of SA and most recommended.

#1 Mark's Outing (closest to the Alamodome)
#2 Mr. Juicy (personal favorite. If they by chance have chicken fried steak (always ask), don't miss it)
#3 Chris Madrid's (most order the Cheddar Cheezy but my fav is the Tostada Burger)
#4 The Cove (great place to take kids)
#5 Sam’s Burger Joint (often has live music in the evening)
#6 Big’z Burger Joint (3 locations: near Seaworld, Six Flags/La Cantera, or Stone Oak near the Temple)
#7 Chunky’s Burgers & More
#8 Smash’d
#9 Burgers By The Butcher at Pullman Market (Pullman Market is worth a visit even if you aren't looking for a burger)
#10 Best Burger

View the full guide here: https://www.expressnews.com/projects/restaurants/best-burgers-san-antonio/