As you say, many hamstring exercises are not very specific to climbing.
Let the acute injury settle, then try the following
1. Bridging.
Double then single leg bridging, the further the heels are from the bum the more the hamstrings are emphasised.
2. Heel drags
Lying on a smooth floor, heels on a towel or similar. Drag your heel towards your body. Progress by reversing the movement: dragging your body towards your heel (the towel then goes under your body, wear climbing shoes for good heel grip).
Further progression: doing the same exercise in sitting: sit on the towel, dig your heel in, and drag yourself towards your heel. This is reasonably specific to heel-hooking.