Check with the hunting associations you belong to. Many have insurance cover, including when hunting. The terms and coverage will differ. I have no idea if they provide cover only when hunting or shooting in the country of origin or not.

Secondly if hunting or shooting overseas, travel insurance MAY cover you, or NOT!? Many standard travel insurance policies will EXCLUDE activities such as hunting or mountain climbing etc. Some, a few, WILL COVER hunting/shooting. PHs in particular often pay a lot more for insurance for this sort of cover, but they are also in the business, which is a different category of insurance.

I believe the travel insurance I took out for my last safari covered sporting (not business related) hunting activities as well as good coverage ie $$$ for loss of my firearms (ie double rifles) if they were lost/stolen etc.

If someone on here can provide guidance and pointers in the right direction, great, but best way is to read the policy yourself, and/or ask the organisation the questions directly.

Liability insurance can be important as it is not only accidental shooting of humans which is disastrous, but also eg accidental shooting of valuable animals eg stud animals. I once shot a couple deer in the middle of a horse racing stud where multi-million dollar racing blood stock was visible in the distance BEHIND the deer. I made extremely sure the bullet went right where it was supposed to, and NO chance of a miss, full penetration etc.