Local newspapers published lists of men from their town who enlisted, especially in the early part of the war. They usually also continued to publish details of local men who were wounded or decorated. I think that the Hamilton local paper was the "Hamilton Advertiser" which may be covered by the online British Newspaper archive or the newspapers on the Find My Past website. Both of those are pay sites, but local libraries often have subscriptions.
Another route is to search the military records on both Ancestry and Find My Past using just the keyword "Hamilton" - there will be a lot to go through, but you will find the records that survive of men who enlisted at Hamilton.
jds1949