For elective rotations, I've rarely seen students spend nearly their entire 4th year at nonaffiliated institutions. Some such schools have a limit of how many months they allow each visiting student to stay, so to remain in one area that entire year, you may have to set up a schedule that utilizes multiple nearby schools.
For audition rotations, practically speaking for most schools, there are mainly late summer and fall months available to try out, make an impression, and get a supportive LOR in the bag that would make a positive impact on your residency application. That application system opens in July, with interviews offered starting in August and generally scheduled for fall.
Because the logistics are difficult, due to outside schools offering rotations on a set schedule unlike your school's (you can't just drop in when it's convenient for you), a lot of required paperwork/getting permission, and interference with residency interviews (a dim view is taken of visiting students taking time off), most students visit nonaffiliated programs for 1-2 months only. Also, be aware that for each school you "visit" for an away rotation, you will have to abide by their individual standards for immunization and titers, which may vary from your home institution's. I've seen some major scrambling (and expense) related to getting this done before deadlines. Hint: Public Health Clinics are your friend.