Check the coil's resistance to make sure it's still good.

https://www.flippers.com/coil-resistance.html
http://pinballmedic.net/coil_chart.html#gottlieb_solenoid_coil_chart

If it's too far off, it may be time to replace the coil.

Is the solenoid not individually fused? Most are on gottliebs... Are you sure the coil is ever actually locking on? If it's not locking in the fuse won't help

I've got the same issue on my BH. Brand new coil, EOS mod, still blows transistors occasionally. I assume it's just too narrowly rated and the coil is slowly killing it. Think I might just mod in a relay driven by the board and be done with it.

That would work as well as the relay, as long as the board's driver transistor can handle it. Not sure how it's specced compared to the under-playfield one

I've seen the pop bumper mod used on BH, so I assume it would also work on HH.

Each mod does slightly different things.

With the the EOS flipper mod and pop bumper mod, there is only a brief momentary firing of the coil, so the coil is theoretically protected from overheating and burning out.

With a relay, there is no control of the duration, so the coil could overheat and burn out, but the transistor would be isolated.

With the EOS flipper mod, you can control the coil strength, but this still does not isolate/protect the transistor.

With the pop bumper mod, you can isolate the transistor, but have no control over the coil strength.

Anyway, most people seem to use the EOS flipper coil mod. If you're still having problems with just the EOS flipper mod, maybe the better approach is to use both a relay and the EOS flipper mod so the driver board can do what it needs to to trigger the relay, and you can use the EOS to control the strength, and the transistor would be isolated from the coil. I'm not sure if anyone has tried that or how successful that might be.

That's my thinking...

Here's the guide for BH. It will likely be a bit different for HH (wiring, etc).

Would it even work on HH? Not sure if there's any place in the up kicker mech to stick one...

I've heard of people trying it (some successful, some not), but I haven't seen photos--granted, I haven't actively sought any out since I don't own the game.