I wonder if using a bigger gun would give higher altitude and let you fire rockets that could go the last stretch on their own?
If I recall, they tried that, in some of the later 'martlets' they tested. They were almost there, and it was always known (literally, IIRC Newton did the original math) that a gun alone could not shoot a projectile into orbit. There's always got to be a second push at the end to circularize the path. Otherwise you either exceed the planet's escape velocity or you have a very high parabola.
Basically though, the budget ran out before they could get a successful launch. Maintaining the guns took a lot of government support because the charges and all the parts were coming from diverted battleship supplies. The grain on the gunpowder such a cannon needs is more like thumb sized plastic tubes, so it's not something you can just go out and buy.
Gerry Bull pretty much alienated his sponsors at the end, got another few months of funding by cooking the books and promising that the site would generate results and become self sufficient, then as a final insult/injury, blackmailed them into selling the site and assets off to a company he controlled by showing up with a court order on behalf of the citizens of barbados ordering the site returned to its original pristine condition whenever operations ended. Since the powers that be had no intention of supporting any further operations, they basically turned it over to him on the pretext that he would keep it running.
Just as a bit of trivia, I'm imagining that made the HARP guns the largest privately owned guns in any collection, at least till the Mossad caught up with Gerry. Right now, they're still rusting away on the beach in barbados.
A