(This is Miriam replying but I'm in Mozzer Alliance right now)
Hi Michael!
I think the most important thing here is for you to give yourself a crash course in Local SEO. This is a good place to start: <cite class="_Rm">https://moz.com/learn/local</cite>
It's vital to understand that your client's local pack rankings are going to relate solely to their physical location. So, if they've got 1 physical location, that means 1 Google My Business listing and 1 set of citations for that location only. You do not want to acquire extra phone numbers or virtual addresses of any kind. Only the real, physical location can be promoted as a contender for Google local pack rankings, unless you live in a very rural area or have a very niche business. Instead, you are in a competitive industry in a large city. So, main thing is to remember that physical address = local pack rankings. All other service cities = organic rankings.
So, yes, you do need a physical address. The phone number doesn't have to be local, but it's considered preferable by some. I don't see any mention of an address on the site right now, so that's something you'll need to address with the client. If the company does not have a dedicated public address, you can use the home address of the business owner and then, when creating the Google My Business page, be sure to choose the service area business settings that signal to Google to keep the address hidden. Regardless, it's advisable to have the complete name, address and phone number (NAP) of the business on the Contact Us page and the website footer.
The site architecture you'll be building out for the moving company will likely consist of basic pages (home, about, contact, etc.) a page for each service (moving, piano moving, what have you) and a page for each major city you serve. The content on these service city landing pages must be unique and of the highest possible quality you can create. This might include text content, video content, images, testimonials, tips, safety warnings, etc. But, again, be sure to warn the client that you do not want to use virtual addresses on these pages or anything like that. Don't be swayed by competitors who are seeming to get away with thin or duplicate landing pages. This is not the way to go and it makes them weak - you can hope to surpass them with a superior effort.
I'm not sure what you're describing about the links/press releases and blocks of cities. Google's Webmaster Guidelines do not want blocks of city names/zip codes anywhere on your website, so I'm not quite sure what it was that you saw.
Finally, yes, you may have a better chance of achieving higher organic rankings in smaller cities more quickly, because the competition is likely to be lower, but achieving this is going to be a combination of time + effort. It's much better to under-promise and over-deliver, specifically as this is your first job.
Hopefully the Moz Local learning center will get you off to a good start, and we have tons of great content in the Local column of the Moz blog. https://moz.com/blog/category/local-seo
Good luck with the project!