Can I apply for outside scholarships (or any scholarship) after my F-1 visa has already been issued?

This statement generally applies to students, who have applied for financial aid, using the FAFSA form.
What it means is that you can’t expect to receive financial aid if your need has already been paid. In other words, it doesn’t apply to you because you are full pay.
If you can find a scholarship, at this late stage, and someone gives you money, you report it to the FA office.