Such a great game for its time. Edit: it's modem <-> modem not LAN so a max of 2 human players. So I assume we were using null modems to play together in the same room.
Seems likely that this Carlton McLawhorn [i] from Wendell NC is the author (and also an avid rocketeer and scoutmaster).
[i] https://stricklandfuneral.com/carlton-carl-bruce-mclawhorn/
There are two parts I remember very clearly. The first is that you have the choice when you leave the house at the beginning of either bringing along a walking stick or something else, and if you don't choose the stick, there's a part on an outdoor trail where you twist your ankle and lose the game.
The other part I remember is getting shrunk somehow and ending up in a drawer full of drinking straws. You have to work your way through this tube maze, but some of the straws are clear (invisible) and if you wander into one accidentally, the game lets you know it was so narrow that you get stuck in it forever, lose the game, and die.
It was a brown-ish DOS platformer, I remember it was distributed (to me) as "mario" even though it was not a Mario clone. Probably EGA graphics? The most similar one I've seen is one of the Dangerous Dave games, but I don't think it was that! It's not "Monuments of Mars" either.
Sometimes I wonder if I dreamed it, or maybe it was a small project someone made and never properly released.
edit - "Rick Dangerous" looks similar too: https://www.youtube.com/watch?v=SNEgrJ9wejo
(For reference: https://www.mobygames.com/game/harpoon )
Edit: I'm now sure that's it. I even found a scan of the original gaming magazine (the German "PC Player") with the test I read back then.
I've been searching for years to find an early multiplayer DOS lan game (~1990) that I believe was called 'Trek'. You separately controlled multiple ships to protect your base and attack others. It was basic ascii graphics but my friends & co-workers spent hours playing via local lan setup.
Hoping it's in this collection!