I don't bother with it on the phone because any time I'm out walking or riding I have a set of dogtags with wife and mom's numbers, plus my driver's license number. (Any cop checking the DL number will get the info that I have a CHL, and thus that there may be a gun somewhere on me or in my stuff that, if I'm not in any condition to answer questions, needs to be secured before the EMTs start cutting clothes off. Wife and mom also know to tell any legitimate emergency personnel where the gun is most likely to be.)

More important would be making sure all the ICE contacts have any relevant information; particularly things like any medications you might be on, (which needs to be updated ASAP when that changes, particularly since that's the most likely time for any problematic side effects to cause the sort of emergency where people might be checking with your contacts) any allergies, or preexisting conditions. If you include your boss or another coworker among those numbers, you might want to seal up a list of the relevant details in an envelope to be opened when an emergency arises, signed across the flap so you can inspect it on occasion to make sure they're not getting nosy.