34048A is the Winchester serial number on the action. If the action was made in 1930, G&H couldn’t have reworked it any earlier than that and by 1930 they were settled into keeping very good records. They were up to about 1000 in their own serial number book.
Is the barrel marked “Winchester”? If so, the rifle was likely only restocked by G&H. If not a full custom build (rebarrel, etc) they were typically not marked with a G&H number. There are rifles that were rebarrelled by them that are marked “rebarrelled by G&H”, but have no serial number.
It’s a complicated story with all kinds of exceptions, but make no mistake by 1930 G&H were numbering their guns