Generally when stitching your overworld areas together, you'd use side warps (Screen->Side Warp). These are up to 4 warps that are triggered when you scroll off one of the four sides of the screen. Here's one such warp in LQftH2 which uses a segmented overworld with multiple different area palettes:
The four boxes in the middle correspond to the directions that are active and which warps they're associated with, in this case only one going to the left using Warp A. Since this is an overworld warp it uses the Scrolling Warp type which imitates regular scrolling behavior between two screens. Beneath the warp type is the DMap it leads to and the screen within that DMap. You can hit Go to test if your warp is linked to the correct screen (handy if you get into DMaps with screen offsets).
The DMaps themselves are set up in the DMap editor (Quest->DMaps). The most important tabs there are the Mechanics tab where you set the map number and continue point, the Appearance tab where you set what palette it will display (under Color), and the Music tab where you set what music to play.
I'm not quite sure what you mean by "after clearing a dungeon" but assuming you mean when exiting via triforce, that also uses side warps. In this case it uses side warp A and you just use the Insta-Warp with Zap Effect warp type instead of a scrolling warp.