My impression is that most tourists stay in Centrum. Some may wander as far as Langelinie for The Little Mermaid or go to Christiania, but other than that it seems to me that they stay within walking distance of Nyhavn/Tivoli.. so the most overlooked gem would be Copenhagen.
Frihedsmuseet, the Museum for Danish Resistance, about Denmark in WWII.
It's excellent and a little unusual (underground and with a cool AV component), and right between Amalienborg and the Little Mermaid. The building is very missable so unless tourists are interested in WWII it's very easily overlooked.
Why the link to a travel website? Do you work for them?
at Jægersborg Dyrehave you have a small community living at Rådvad you can just walk into. As a non-danish, the park itself is amazing but walking through that place on a warm summer day feels like discovering a hidden gem.
My impression is that most tourists stay in Centrum. Some may wander as far as Langelinie for The Little Mermaid or go to Christiania, but other than that it seems to me that they stay within walking distance of Nyhavn/Tivoli.. so the most overlooked gem would be Copenhagen.