Sadly, Jayne, without a starter, many of us never get much further than you have. The usual starter is a "regiment" and "regimental" number and they can be incredibly elusive.
You may have tried but you should use the on-line newspapers to see if there is any report of JAH - no guarantee that you'll get a number there, of course, but you may find important things.
The other thing is to see whether there are any Absent Voters' Lists for the area where he lived. (See http://www.longlongtrail.co.uk/soldiers/how-to-research-a-soldier/finding-soldiers-through-the-1918-absent-voters-lists/
). These documented the (in effect) postal votes that people were given at the end of the war. Unfortunately, they don't always survive, but if they do, they might be the only evidence of someone's number. We knew that my maternal granddad served in the Cheshires (like you, from a photo) but since his name was John Griffiths, we had a wide choice! Fortunately, the AVLs for Cheshire are pretty good and this gave his unit and number. Paternal grandpa, on the other hand, seemed never to have applied for his vote until back home - but I had other evidence there.
I don't know if JAH lived in Scotland but I have no idea where the Scottish AVLs (if any) might be. (In fact, I don't know where the ordinary electoral registers for Scotland are).
See also http://www.longlongtrail.co.uk/soldiers/how-to-research-a-soldier/