In my own teaching experience, I allow overloads, or extra students, only if the student has put the course off until the semester of graduation (which happens with students often in my subject, mathematics).
If your college or university so offers, you may also ask if your institution has "independent study" credit. In this case - especially if all seats in the lecture room remained filled until the first exam - you would be expected to work on your own apart from prearranged office meetings with the professor. This path has the advantage of providing compensation (however small) for the professor taking the extra load.
As a last resort, your institution should have a waitlist for each of the filled sections. Get on all of these waitlists, especially if the course has a high drop/non-participation rate that shows up early on (at my institution, I'll lose three or four seats right away in sections of our higher-level stats course. If you get in before the last add date, get prepped to do a week or so of catching up right away.
If all available paths fail, you may do well in purchasing the material yourself and preparing for the next session. Auditing policy may not allow for this, but usually seats start to empty after the first few weeks (again, especially after the first exam) and in my own experience I have not found occasional students attending the class to be disruptive or against institutional policy.