Spongy subfloor under hardwood

[Cruiser92]

Having loads of squeaks in my hallway, I've removed the old hardwood thinking all I had to do was to lay new boards down. Have found that the subfloor is thin plywood that sags between joists and causes the squeaking. Is removing and replacing the subfloor warranted or would inserting long screws into the joists through the existing plywood be sufficient? Am not sure if floor support is an issue with hardwood floors.