Is there any specific way that you can tell (become aware) that you've entered the Twilight Zone?
It depends. According to the opening narrations in the classic serial, that there were a number of ways to identify if you were in the Twilight zone,
Season 1 - [If] you've entered a dimension not only of sight and sound but of mind. A journey into a wondrous land of imagination then you're in the Twilight Zone.
Season 3 - if you see a "signpost up ahead — your next stop is the Twilight Zone!"
Season 4 - "[If] You're moving into a land of both shadow and substance, of things and ideas then you're in the Twilight Zone.
Seasons 3&4 "If you unlock a door with the key of imagination then you're in the Twilight Zone"