Catrin, you can't improve if you don't have many places to practice what you want to do!
Besides, 6 mph is a perfectly respectable speed for steep climbs. Unless you are racing, what's the point of going so fast that you eventually blow up and have to get off the bike? I have seen this time and time again on rides; people are mashing up the hill, trying to go faster, when suddenly, I notice it looks like they have been shot backwards out of a cannon. Get in your granny gear at the bottom of the hill and stay in a mid to higher range gear on the back. Shift down as you need to on the rear, trying to keep a steady cadence. Do *not* look at your computer! I have been told time and time again that I am a good climber... but really, all I do is use the easiest gear I can, spin up the hill, and pace myself.
The best thing that ever happened to me was on our recent trip to Spain, they did not have computers on the bikes. At first, I complained. But, as we got into the "mountains," I appreciated it. DH and I just ticked the pedals over again and again. We had 3 very difficult climbs (two of them ranked climbs from the Vuelta and one 10 mile climb), as well as a few others. We made it up all of them ahead of everyone else in the group, except for the 2 people who were clearly stronger, who were trying to go as fast as possible. After riding at home for the past 6 weeks since we got back, I now realize we must have been going between 3-6 mph
. I think this is especially important if you don't know what lies ahead.
Don't worry so much about this. It will come.
2015 Trek Silque SSL
Specialized Oura
2011 Guru Praemio
Specialized Oura
2017 Specialized Ariel Sport