In what I consider to be “new school” D&D, movement outside of combat is often fluid and happens via negotiated narrative rather than explicit rules, whereas movement within combat is heavily systematized and makes use of explicit, quantified geometry using a grid. Options are strictly regulated by action type. Improvisation, based on referee ruling, is of course still possible, though the outcome of such ad hoc rulings must ultimately be represented within the geometric structure. Time is usually kept strictly within combat, but not outside of combat.
Old school D&D, in practice, often represents movement and action possibilities both inside and outside of combat via negotiated narrative and without much strict numerical quantification. PCs move through a hallway, into a room, enter combat, charge, retreat, and so forth, all without direct geometric reasoning. However, the older rulesets, as written, have a focus on logistics as seen in movement rates by armor type, chance of random encounter per turn of movement, light radii, escape, and other similar rules. This seems to gesture toward an almost inverted type of game geometry compared to the new school: heavily quantified movement outside of combat with narrative, fluid movement within combat.
Following this structure, old school D&D could be seen as more of a board game outside of encounters whereas new school D&D is more of a board game inside of encounters. Comparing aspects of RPGs to board games is often seen as pejorative, but such is not the intent here. Instead, I think it might be interesting to try taking this board game aspect of old school exploration movement more seriously. For example, a scouting, unarmored, stealthy thief, with a movement rate of 12 squares per turn, suddenly becomes hugely valuable in the sense of quantifiably being able to cover more ground per exploration turn. While this might be easier to do in person, with something like a dry-erase battle mat for dungeon movement, it might also be possible online using a shared whiteboard such as Twiddla or Awwapp. This approach could also be taken to wilderness exploration using an explicit player-facing hex map (which I have seen discussed before, but never seen in practice).