Yes, but only because they deferred on producing more Henleys and Hotspurs. Hawker went into overdrive to build the Hurricane, as I said, but quick production, such as that during the Battle of Britain didn't just 'happen'. High production figures British firms experienced during the war were as a result of the war. working hours became longer, more factories were built, Beaverbrook in May 1940 threatened production managers, etc. None of this was gonna happen beforehand.
You've identified pretty much exactly why the Hurricane didn't enter service faster, but not why it wasn't produced faster - it simply couldn't be. That production increased was because of the war.
We have a tendency on this forum of just assuming that firms were going to do exactly what the government wanted, such as Fairey being asked to build Spitfires for the Admiralty, and like Fairey, it was another firm's prerogative to refuse and in the pre-war environment that is pretty much what happened. Hawker owned Gloster, which is why it could get that firm to build Hurricanes. Companies like Short Brothers had a government stake, so were more amenable to building other firms' aircraft, but if, say, Westland said, "nope, we're not building Hurricanes", there's not a damn thing the government could do about it, apart from threatening to not give them any more work. This of course happened; Boulton Paul wanted to build better Defiants and derivatives of the design, such as a single-seater, or even a naval Defiant, but the government said "no, you're building Rocs." So, they built Rocs. They could have refused, but it is bad practise to refuse a government contract, but pre-war, it happened if the firm had work already.
Again, how? Ending Gladiator production, i.e. not issuing any more contracts and focussing on the Hurricane was done, but stopping a production line isn't that easy, so finishing the work started was of utmost importance.
The Castle Bromwich factory which historically produced Spitfires was under construction in 1937, with no Spitfires that would be a Hurricane factory, setting up production is quicker with the Hurricane and it needs fewer man hours to produce each plane. Hawkers were the only big players, they had an interest in a lot of pies, like AVRO for instance.