Wilderness as a dungeon

Here's a thought: Treat city streets as corridors of a dungeon that players must move through and map. Of course, since they're outdoors they move at a faster rate - 10 yards per turn of mapping, or 100 yards per turn not mapping.  They can accurately map out a city, and the DM can explain it, by walking down the streets and noting intersections and important buildings along the sides. Of course no one does it this way, because it's too cumbersome.

Traps in the wilderness are way more natural to my imagination. Snare traps hidden under a pile of leaves, pits dug into the ground and covered with moss, spikes dug into the ground, nets in tree canopies, razor wire across barks, bear traps in the ground. These are all great traps that have actually been used in the real world. It's gruesome but the fighting in Vietnam and WW2 in the Pacific have many examples that can be used for inspiration.

Wandering monsters actually have a reason to wander in the wilderness. In a dungeon, it's assumed that they're aimlessly walking about, or occasionally stalking the players. In the wilderness, they could just be going somewhere. In the case of orcs and goblins, they can actually be an organized band on patrol. Maybe they're actually a unit marching to a destination and run into the players by accident. Maybe they're camping and taking a rest. Maybe they found the players' tracks, and are actively following the players. Maybe the players accidentally walked into the territory of a large predator. All of these create encounters with a purpose.

Wilderness travel, for small scale exploration or town activities, should be measured in terms of hours. The conversion is simple, 60 yards for every hour of travel in an unknown wilderness, if mapping carefully, so a character with 12" movement can cover 720 yards per hour.
Without mapping, that's 7200 yards per hour, or roughly 4.09 miles per hour, which is about average human walking speed IRL.

Also, moving over the hex map - Give the players a large radius of sight, but don't reveal the whole map. Have the players move with natural directions - orienting towards a landmark, or a compass direction, and setting a distance. Reveal hexes according to the players' sight radius, or multiple hexes at once. Obviously this technique is meant more for VTT that handles line of sight for you.