Yes, the shock from the hammer will jar the garter spring loose. Real seal install tools have a tube the expands the lip so the spring won't pop out. ( same dia as the shaft to be installed )

Without a seal tool, packing the back with grease keeps the spring from pooping out.

To fix, get some paperclips or slightly stronger wire, bend one end into a 90 a space then almost a 90 again. The spacing should allow the hook to slide over the outside of the seal, down the face then under the lip. Make 2 of these.

Take a third and bend it into a nearly 180* .

The goal is to use the first 2 of these clips held at 90* angles to keep the spring in place as a the third works the spring around. The spring won't stretch as much as you will be folding the lip in slightly. The end of this 180* wire will push the spring, the seal lip will be in the V of the 180* bend, this way you can stretch the spring if you go in one direction and pull the lip to the center by pushing in the opposite direction.

You may have to re bend the paperclips / and or have a second person hold the stationary wires.

The process does not require great force, it is a exercise in balance and finesse. Go to the bathroom _before_ starting this process as it reduces tension. ( really, it works great for removing auto glass too and no, don't go on the car )

Some front transmisson seals ( GM ) have a lip that curls back around to prevent the garter spring from popping out.