About 40 years ago, My GPA was very low (about 2.2), (However I did make a 3.35 for the last semester). I worked 40 hours a week at a non-professional job. I was upfront. I went to a professor teaching methods of mathematical physics (graduate level not undergraduate), and asked him if I could take his course as a "Special non-degree seeking student", (I already graduated with a BS. Physics) I also asked if I did well could I get a Letter of Recommendation from him. He agreed.
I was financially unsupported by the school. My job gave me a living. I got a B+, applied to grad schools in the conventional manner, (scored about 60 percentile in the GRE), got 3 letters of recommendation etc. In addition, I visited one of the prospective grad schools, met the faculty. Asked intelligent questions tried to be very involved in their research areas and I think I impressed them.
Bottom line. I got into grad school (a respectable one) the next year with only taking that one graduate course. Actually when talking to faculty a few years later about my case, they told me (surprisingly?), "Why wouldn't we take you." Ultimately, I was successful in grad school too.
At least as of 1979, it was possible to go from special status to grad school as long as you apply in the conventional manner, with test scores and show a strongly improving trend.