The matchbook Jim shows is actually Wimpy's #2. Built in the mid 1930's on Fremont between 2nd (now Casino Center Blvd) and 3rd. It was somewhere in the area of where the Fremont/4Queens now stand.
The original Wimpy's (#1, early 1930's) was an itty-bitty, little wooden shack at Fremont and 3rd. It had no dining room, only a window to hand food out of. Maybe big enough for 2 employees and needed equipment...You'd know it now as the East end of the Fremont Experience where the tiny, old "gift" shops are....so the brick building (#2) was a big step up from the original location (#1).
I believe Wimpy's closed around 1943 and the owners of Wimpy's built a huge drive-in between Downtown and the Strip on Las Vegas Blvd called "Sill's Drive-In". I THINK I remember eating hamburgers in the back seat of our 1959 Cadillac (aka: The Sherman Tank) in the Sill's lot around 1962 with my parents and screaming little brother (I just remember a huge drive-in near downtown, so I'm guessing that it was Sill's).
My dad always got silly and would say "I'll gladly pay you tomorrow for a hamburger today".
I remember they had yummy strawberry milkshakes, a BIG treat for me to have one!
Sill's eventually burned down...I think my dad cried.

Yes, In-N-Out's "double-double, animal style" is my favorite...but Binion's Snack bar really does make a great, juicy fresh cheeseburger. Great, now I'm hungry and it's bedtime!