Hi Activity Super,
You are quite correct that this is a challenge faced by countless businesses just outside the borders of major towns and cities. Because of Google's stance that a business is most relevant to its physical location, you will typically be seen and displayed as relevant to your town in the main organic/blended/local results - not to a desired nearby major city. That's just the way it works.
An exception to this is when you offer a service that is relatively unique to a large geographic area (the only large animal vet serving 4 small towns) in which case, you may find yourself dominating the local rankings for a variety of geographic terms due to lack of competition.
Because of the way Local works, then, most businesses shoot for secondary organic rankings (rather than local ones) in towns where they do not have a physical locale.
What is your business model? Are you actually an ironing service? Do you serve all clients at your shop or do you travel to some/all clients to render services? Defining this is critical to the handling of your Google Place Page.
If all clients come to you, you can show your address in G. Maps and attempt to use the old technique of making one of the additional details on the Place Page 'Areas Served'. In this, you could list up to 5 cities (do not include your city of location) from which clients come to you. I refer to this as an old technique because it is one that worked well prior to Google's layout change last July at which time they stopped publicly displaying additional details on the Place Page (as well as citations and other elements). Though Google is no longer displaying this information, they are still aware of what you put in the additional details. The thing about this is, I'm not sure if this technique works as well as it did prior to the change. Still, it wouldn't hurt you to try it.
If your business model serves some customers in your shop, and some on the road, you can show your address and also choose the service area radius feature, in which case, you could include Essex in your radius. This could be of modest help.
If your business model goes to all customers and does not do business at your physical location at all, you simply have to hide your address.
So, that takes care of the Place Page.
Other local business directories often offer you a much larger field for describing your business. You could certainly work Essex in there somewhere.
What you are left with beyond this is writing up a storm about Essex and how it relates to your business. If you've got a go-to-client business model (like a chimney sweep) this is a no-brainer, because you can simply write up your work in that city. If you don't travel to the town, however, you have to be more creative and become involved in some way in Essex in order to have something to write about. For example, a physician might give a lecture or a clinic in Essex. A brick-and-mortar business might host an event in Essex, or sponsor a children's sports team there and attend matches. Perhaps there are issues that uniquely effect your Essex-based customers (such as laws, weather, special offers, etc.) that you could write about. The thing is to find some genuine reason to connect your business to the major city in the absence of actually serving clients there.
Then, as you've discerned, linkbuilding is step two. Some people engage in article writing as well, but in my opinion, investment of time in this should typically be minimal. It would be great if you could get indexed news about your business from Essex-based news sources but this isn't always possible. Also, you might play with using hReview to highlight testimonials from Essex-based customers on your website...not sure what impact this would have, but it's an idea.
Essentially, then, a combination of a properly optimized Place Page, other local business listings, copywriting and linkbuilding will be your best bet for breaking into next-door city rankings, but if you are in a competitive industry, these rankings will typically be organic rather than truly local. Some visibility is always preferable to none!
Good Luck!