@Malcat Thank you! Approaching it as "I already have a project, can I work for you" sounds a lot better than "can I be a free intern?"
It was more that I was already networking with tons of businesses through my executive non-profit work, so my value to these companies was already established, and I had excellent relationships with these people already, so it was an organic process of exploring our potential value to one another, and trying things out to see if they would work.
However, yes, I have just cold called organizations and offered to help them with something, or try a pilot project, etc. I've either done things entirely unpaid, or with a clause that says that if we decide to continue moving forward with our work, that I get back-paid for my time. I was almost always an independent contractor, so it was set up basically that if they were satisfied with my provided service that I got paid and if they weren't then I didn't.
There are countless different ways to leverage unpaid work, but high level networking skills will make it a much more powerful lever.
FTR, this is easiest with small and medium sized businesses where it's easier to get direct access to the people who make executive decisions, and where there is more flexibility in creating new roles. Also, smaller businesses have so few staff that it's far more common that they will have a lack of specific skills among their staff, just by virtue of having so few. So it's easier to find a business that could really use technical services, or writing services, or help with logistics, or health and safety, or financial advice, or business development and marketing, or management and training, etc, etc, etc.
Almost every small to medium business I've ever advised could desperately use some kind of consultant for some kind of service, but consultants tend to be very risky because they're expensive and give unpredictable results. Most of my opportunities came up just from shooting the shit with other professionals about how brutal it is managing businesses, and figuring out how to be useful to them.