How can a marketing leader at an early-stage startup effectively gather insights from new market segments when the company has limited resources and no existing relationships with potential customers in those segments?

Based on your budget, ranging from zero to small, I'd consider the following options:

Zero Budget: I'd do a combination of desk research on the industry and listening in on prospecting calls from SDRs and seeing how those prospects in different industries resonate with your messaging/describe their problems.

Limited Budget: I'd try something scrappy like building landing pages on my site and then creating highly targeted SEM campaigns with a fixed budget so that I could see what segments converted and showed a signal that can be used as a proxy for interest.

Small Budget: I'd try something like the targeting features within a platform like Survey Monkey or try to self recruit for a small focus group.