How likely are Veterans be to granted a scholarship after exhausting GI Bill benefits? Which ones?

As a veteran who also exhausted their GI bill on my undergraduate and went on to do multiple graduate degrees, I completely agree that it entirely depends on you and what career path you are pursuing. My military experience has always been held highly favourably, including in appointments outside of the US (Europe and Canada), as you are always assumed to be more mature, more able to handle pressure, more likely to stay task focused, and frankly more likely to show appropriate deference and loyalty to supervisors without being a complete pushover. As far as scholarships, I stayed in science, and as such have had all of my graduate degrees entirely funded + stipend with very little issue. The biggest issues have been on my end, as the wishy-washy academic world can be massively grating when you’re used to a clear (ish) mission and expectations as to how you will perform.