I've been on both sides of the fence, and hired from academia into industry.
There is a lot of variability, of course, but generally references matter less and impression made during the interview a lot more for jobs in industry vs academia. Accomplishments and credentials continue to matter.
In industry, many people who change jobs are doing so since their previous job is not a great fit. Or have not let their current manager know they are looking elsewhere, so cannot use them as a reference. And getting a good reference from someone else is easily gameable. As a result, many reference checks -- at entry level in particular -- are purely formal ones, merely to make sure the candidate isn't a conficted fraudster or lying through their teeth about their prior experience.
To the extent you are applying for more senior and/or creative positions, there may be a desire to genuinely talk to someone to get a better sense of what kind of person you are. So you should think through what articulate and ideally industry-experienced contact, another faculty member or even a fellow research team member, you could supply if someone does want a reference.
By the way, we of course can't know about the specifics of your relationship with your advisor. But it is certainly possible that a dispassionate (as opposed to loyal) advisor may have given poor references during your academic job hunt not out of personal animus, but since they were not convinced (rightly or wrongly) that you are well suited to continuing in academia. They might be quite supportive of you seeking employment outside academia and even able to provide a good reference in that situation. Of course, this may not be the case at all, especially if you feel you cannot trust your advisor to not be two-faced.