Shift items to another closet entirely. Unpack hanging closet storage racks and multiple garment hangers, and hang each item individually. This distributes the weight more evenly across the closet rod. Or at the very least, hang heavy storage rocks near the ends of the rod where it’s less likely to sag. Stack shoes on the closet floor, or store them in an over-the-door holder, instead of in a holder suspended from the closet rod. Store out-of-season clothes, especially heavy sweaters, under the bed or on closet shelves instead of hanging from the closet rod. Sort through your closet and donate unwanted clothing to friends or a charity; this might lessen the stress on the closet rod.
The easiest way to do the latter is to measure how long the current brackets are.
If you don’t have a stud finder, tap gently across the wall. The drywall will sound hollow between studs, and firmer as you’re knocking over the stud itself.