Ideally, we would like to update to 5.7 which we know does require a new installation. The one thing we want to be certain about is that it does not mess up our SEO.
so long as you don't have any downtime, your server isn't loads slower and you keep the same url structure and your content is the same you should be ok from a SEO point of view.
There is no direct upgrade from 5.6 to 5.7 but there are 'some' migration routines from the latest version of 5.6 to 5.7 which might help if you have lots of page content, page templates etc.
However from 22.214.171.124 (if memory serves me correctly) you'll need to step the upgrades through 5.5 to 5.6 - 126.96.36.199.
In your situation I'd be looking at the site to judge if it is worth attempting to migrate or just build a new site in 5.7, create the same structure and just copy the content in, spider the site, check all the titles, content and metadata etc before switching over. That all depends very much on how much effort either approach would take.
You could update to the latest 5.6 version and a mobile design. Would that address your issues?
My name is Jordan Santaga and I've worked as a web developer for the past 6 years, specializing in Concrete5 powered websites. If you are still looking for a developer to help you with this upgrade I would love to be of service to you and your project.
I too might recommend just re-installing the site and re-building the same site structure and content using the updated theme but if you wanted to incrementally upgrade that is an option as well. Please view my C5 website to learn more and see examples of my work:http://www.trinamicdigital.com
Feel free to message me with any interest you may have.