Both times it was as a contractor. The first time I was employed by a body shop as just simple staff augmentation. This was in the 2000s, but I believe they are still around. I think if I'd wanted to convert to full time it would have been a process, but I definitely got to meet employees and managers from the bigco (and connect to them on linkedin).
The second time was in the mid 2010s when a friend was at a big company (his startup had been acquired and then acquired again). The friend was able to intro me directly to the hiring manager's manager. We had a friendly conversation, I had a quick tech screen and they brought me in. That would have been no trouble at all to convert to FTE (in fact they had a conversation with me that began with, paraphrasing: "how much do you like money?").
I think the best way into a larger company as a generalist beyond the early stage of their career is an acquihire, but the second best is as a contractor. Risk is lower all around (well, except for you).
When some of the contractors went FTE at a previous role, their salaries dropped dramatically (though they were paying for their own health insurance through an agency).
Nothing wrong with smaller companies and startups, but the pay increase and resources available in a larger company are something I would like to experience. Getting my foot in the door is a frustration though.