I think that the question of WHO to call with a situation is more important than how.
The how is simple. A sat phone. Ham is down the list of how.
Now the question of who.
You have to remember that sat phones are down linked only in a few places across the world. So dialing 911 may or may not be effective.
You need to have the local numbers for emergency services. for the locations you are going to be in.
IF you are traveling abroad, you also need the numbers for the US consulate, and the state department back in the US. You may run into a situation that you have issues with the local or state government where you are and the state department is your only real assistance.
You may also contact those entities prior to travel and ask them if they are monitoring any HF / VHF frequencies that can be used in an emergency. Remember that you are a US citizen. (Unless you aren't but this concept would apply, the contacts are just different).
But on a side note. Having the ability to communicate with the US government in another country that's not so friendly to the USA may well get you labeled as a spy and raise MORE problems.
All that being said. If you are just running around Montana, Wyoming and other remote locations in the US, get a sat phone. Put the emergency numbers in it for the local public safety folks and call it a day.
If you are dead set against that. At minimum carry a GPS, and those phone numbers with you. That way when you start hollaring for help on 75 meters once someone in Ohio answers you, you have the information to give them with the proper contact information for them to make a call so you might actually get rescued and not recovered.