It seems to me not all state militias and mounted units went to serve outside their states during the Mexican War. From: "New York State Military Records - Before the Civil War" http://www.nysl.nysed.gov/genealogy/precivil.htmhttp://www.nysl.nysed.gov/genealogy/precivil.htm
New York State furnished two regiments of Volunteers: The First (later renumbered 2nd) and the Seventh (later the 1st). The service records of these soldiers are on file at the National Archives and Records Administration, Washington, DC 20408. (The First fought in Mexico; the Seventh served in California)."
The unit mentioned is the 3rd Regiment so it seems to be one unit that stayed home in NY and cannot be given the distinction of "Mexican War Veteran."
Here is a pretty comprehensive database of all officers that served in Mexican War and Levi Speelman is not included nor is Col Thomas Hall. I did find a George B Hall, Capt USV (United States Volunteer) 2nd NY Vols http://www.aztecclub.com/search1.htmhttp://www.aztecclub.com/search1.htm
I find his bio in a Yates County NY History and it does not even mention any military service and says he he was born in and never resided outside of Yates County his entire life.