If it was a very faint H, it would not have been worn over time in the case of your coin because the date numbers would have been in higher relief compared to the H, this means the date would've worn down before the 'faint H'. As the numbers on your coin are not as worn, there would still be a faint H present.
However, it is possible as many have suggested, that the H on the die became completely filled and struck coins with space under the date completely flat to the field. Whether or not this will show up an outline of an H on a high powered microscope is something someone needs to check, but it would be far better with a higher grade example to be more conclusive.