The Incredibles seems to mix modern technology with the style of the 60s.
Isn’t this pretty much what happened on Mad Men too? There was an episode where Roger took LSD at a posh party before it was a thing. (Timothy Leary started promoting its use around 1963.)
Roy Rogers was billed as a “contemporary western.” Such settings were still around in the 1950s, but the genre died out as they became increasingly improbable.
From what I remember about Sky King, it was pretty much the same.
IIRC, the first season was very sloppy about anachronisms. But after the show went viral with people picking at every little detail they got better about sticking to the proper time frame. Until the last season, I think, when they just wanted to end it.
That was season five when the show was well within 1966. Roger was a late comer to acid compared to the Manhattan intellectuals he was hanging out with.
Growing up, I watched a lot of TV shows set during WWII (e.g., Combat! and 12 O’Clock High). Whenever a female character appeared, my mother would say something like “Holy 1967 hairdo!” 
Yeah, I remember they were really into it. But I got the impression it hadn’t caught on with the counter culture yet.
LSD was never really a “street drug”, was it? It started out as a psychiatric medication. It makes sense that upper-class libertines would be using it before the hippies discovered it.
Timothy Leary created Millbrook House in 1963 after his experiments with LSD on patients (and himself) saw some success. He proselytized the drug from there for 5 years until it was declared a Schedule 1 drug in the U.S., making it essentially illegal and hard even to get medical justification for experiments.
Owlsly Stanley was the one who turned it into something close to a street drug, when he manufactured 300,000 doses in 1965, and 5,000,000 more by 1967. His association with the Grateful Dead pushed the drug into SF’s hip culture. They created the hippies, arguably.
Maybe there were groups in every large city or college town who were partying with LSD in the early 60s, but getting a supply would be hard almost everywhere.
John Lennon got it from his dentist. Was he an “upper-class libertine”? I’d need to research that.
The Merry Pranksters were handing out LSD to anyone interested by 1964. “Acid Tests” were at their height in 1966, so Mad Men pegged it pretty precisely.
Gattaca has genetically engineered humans making daily rocket flights. But everyone seems to be driving around in 1960’s cars. Especially Rover P6s.
Black Mirror also seems to like old cars in the future.
And Hello, Tomorrow! very much looks like the 1950s. The fashion, architecture, cars, etc. all look like they’re from mid-century America. But they have all sorts of whizz bang futuristic technology – robot waiters, automated delivery bots, regular space flight, hover cars (but they still look like 1950s cars, except they hover). I’ve seen it described as what people in the 1950s imagined the future would be like.
Yeah it is amazing how in Outlander she found the only period Scotsman with perfect teeth, an understandable accent, and a progressive viewpoint. But as you say, sometimes a little accuracy has to get out of the way of a good story. I only quibble a little, but it is like showing the srat actor in a medieval pic with his helmet off so we can see him act. Fair in the case of art.
Roy Rogers kinda made sense, but having every mystery be solved by having a airfield right there to land the plane was a bit much after a season. But four seasons plus a radio show?
Same with Star Trek TOS.
The “Beyond the Sea” episode seems to be set in the 1960s, but has a space station and consciousness transfer between astronauts on the station and on Earth.
Never watched that one. But I was happy when one wild west comic I read had a telephone. Because some businesses did have them in the 1880s and 1890s, which a common setting for wild west tales. Electricity makes me happy for those, too (I’ve read old newspapers, and even small cities out west tended to get electricity sooner - they were new cities that didn’t already have gas infrastructure).
I know of two different comic book western characters that started out as contemporary, but then shifted to the 19th century without comment.
Another show possibly in the same universe, or at least from related source material is Netflix’s Chilling Adventures of Sabrina, which has a similar mix of modern and 50s/60s elements. Some of that may be deliberate characterization, because the long lived witches may have just not bothered to update styles, cars, and views in 60 years, while the mortals live in 2020.
If sci-fi counts, then the Fallout show and games has an odd setting. It takes place a few hundred years (or something) after a nuclear war that happened during the Cold War, so everything has a 50s/60s vibe, but the technology at the time of the war is far in advance of what we have now. So the war happened both in our past and in our future.
Disenchantment starts off in the kingdom of Dreamland in some undefined fantasy medieval time, then we run into the Victorian-ish Steamland, with its steam-powered airships and electricity and robots and Tel-O-Visors.
Not to mention all the pirates and mermaids and elves.
Really, the stience is all over the place.
The Hudsucker Proxy nominally takes place in 1959, but everything looks more 1930s-ish.
Blackadder is an example of this. I feel it’s most notable in the third series. Historical events which occurred during the series include Samuel Johnson publishing his dictionary (1755), Pitt the Younger’s election (1783), the French Revolution’s Reign of Terror (1794), and the Battle of Trafalgar (1805).