It's definitely interesting -- what I'd consider some of the more well-known and high profile "giant monster" movies like THEM! (1954) or "It Came From Beneath the Sea" (1955) were done in the early to mid-1950s. Many of the pre-hero giant monster ideas were clearly lifted from some of these movies and similar stories -- but they came 5 to 6 years after some of the more popular giant creature movies. But remember 1955/56 launched Godzilla, so it's possible that the comics were just a few years late in picking up on this concept, but once they did there was no stopping them! Gorgo which was a Godzilla clone, was a 1960 movie, so maybe the genre was just a late-bloomer for comics?
In addition other things happening in the late 1950s early 1960s were flying saucer sightings like the very famous Pacific Sightings in 1959, on the heels of Roswell NM in the late 1940s, etc -- and a lot of the pre-hero stories reflect the concerns of alien invasions, giant creatures (many of them extra-terrestrial in origin), and witchcraft. In 1957 the Russians launched Sputnik, and the whole Red Menace issue took on new life.
So you have a LOT of things coming together in the late 1950s, early 1960s that stoked the imaginations of the public (and Stan Lee!) at the time. And thank goodness they did or we would not have had these wonderful comics!
Good questions, especially when you compare S-F vs Monster era :
TOS #1 (January 1959) - #4 (July 1959) : S-F
TTA # 1 (January 1959) - #4 (July 1959) : S-F
ST # 67 (February 1959) - #69 (June 1959) : S-F
Then starting in August/September 1959 the monsters start in the three titles
Could it be linked to some movie (or movie genre) that became popular in the summer of 1959 ?
Searching on the web I found the following movies from year 1959 :"Giant Gila Monster", "Behemoth, the Sea Monster", "Gigantis, the Fire Monster" and "Monsters of Piedra Blanca" that have the same type of monsters than the one seen in ST, TOS and TTA from 1959 on...
Before 1959, big monsters movies were more S-F type. Maybe that's where Stan Lee got his inspiration.