Best approach to ranking new location for a local business

A little background first to what I have done so far. I have been SEOing a local business with 1 location and I have their homepage ranking on the first page of Google for the various combinations of board/general keywords for their type of business + their location (i.e. dentist in city, city dentistry, dental service in city, etc..) and also their internal pages with each page focusing on specific service that they offer(i.e. teeth whitening city, tooth extraction city, cosmetic dentistry city, etc.).

Each of these 6 internal pages are focused on one specific service and it has been optimized for the variation of keywords for that service + their main city (i.e. teeth whitening in city, city teeth whitening, teeth whitening service in city, etc.). The city is also used in the title tags and meta tags. Also it is used 1 time within the content of the page. This same city is also used in a similar way for the homepage content (title tags, meta tags, 1 time within the content).

On the home page we have their phone# and address of their location in the header, a list of the various services with a link to each of those pages as well as a navigation menu on top for contact us page, about us, blog, etc.. On the footer of the homepage and internal pages we have a 1 sentence that describes their service in their main city and also lists the other neighboring cities that they service.

So at first we focused on getting their home page and 6 internal pages ranking for their respective broad/general terms + city combinations (homepage) and also the specific service pages with their keyword/city combinations. Then once we got solid rankings for those we went after the next local city that they serviced and started to rank the 6 internal pages for that city and so on. We didn't change any content on these pages except for just having that city listed in the footer.

But now they have opened a new office that is like 1 hour away in a different county and new surrounding cities. They want to start ranking for this new location (and surrounding cities) with the same website.

So what I was thinking of doing is adding a new page titled "Locations" or "Offices" (i.e. domain.com/offices) (add this to the menu navigation on the homepage as well) and then on this "Offices" page now list the two 2 locations that they serve with just the address, phone#, map, hours, etc. Then hyperlink the new location to a new page that I will create that will have content (300-500 words) about the specific location of this new office, describe all the services that they offer (give brief overview from the 6 service pages, but have it all in one page), possibly add a few images. etc. So this new page will be sort of a summary of the 6 service pages (within a link going back to each of the service pages) plus incorporating the new city in the content as well as mention a few of the surrounding cities as well. Then I would build backlinks to this new page using the various service keyword + new city combinations. Hopefully we start to see this new page to start ranking and the link juice from the established domain helping it along the way. This way I keep all my content the same and keep the rankings that I have already got and start working on the new page to it ranking for it's new city.

So will this strategy work fine in my situation? By just giving a brief overview of each of the specific services (and not a 1 whole page dedicated to each one) and using the new city a few times in the content will that be more than enough for onpage optimization and then I would just build backlinks using all the keyword combinations for this new page? Anyone have any other suggestions or feedback that might work better or be used instead?