Another tip is to develop a fast power-walk... you’ll be surprised how much in races this will be needed... I can think of at least one Shropshire race where none of the climbs are runnable, except for the very best few.
You’ll need it a lot, and as well as building an efficient and comfortable technique, it’s also great work for the lungs and legs. Probably over 50% of my hill sessions are done at a fast power walk rather than full out running... this may be to my detriment, but find I can do day after day of this, and I’d class myself as a good climber (well much better than I can descend anyway).
If possible I’d also recommend some squats. I do these at the end of a session, and don’t “lift heavy”... possibly 20 or 30kg, or 50kg if I use the seated press. Again day after day of this seems to have given me a reasonable engine...
While I’m on a roll, I’d also recommend some quality sessions on the flat. Since I joined my local track/road/cross-country club, my performance has really increased (by my own meagre standards anyway).