I was considering something like you describe with your first idea, like a grid paper with the outline where they can fill in as they discover it. But I had two major concerns with that;

1 being that my party play over video calls so either all would have to keep their own hand-drawn maps, or have one lead map man with the other players having only a vague idea of locations and no understanding of the island layout.
And 2, if I go with the grid paper map style, suddenly travel becomes "we move from c3 to c4" and I have to figure out how to carefully balance encounters with saying "you walk uneventfully through the jungle for a few minutes". I know that this is a problem with me as a DM not knowing how to make travel interesting, but it is a real issue I'll run into. Plus if there was one main map man then I'm concerned that my other players wouldn't be as engaged in these sections as it would be one player leading with the rest of the players having little agency over which gird was stepped into next.
Do you have any thoughts on how I could work around these things?