Local, or map search, SEO has emerged as a distinct form of SEO over the last few years. Up to 4 customers out of ten, on average, now find local companies through maps. As such, strong map visibility is crucial. Instead of a website, local SEO requires that you optimize your Google My Business (GMB) profile with relevant text and links. The strategies for improving local SEO include NAP (name, address, phone number) consistency, link-building, image geotagging, and more features. Ensuring consistency and giving the appropriate time and energy to local listings is a critical factor for getting free traffic.
Redesigning a site has a lot of challenges for keeping the earned organic traffic. A faulty redesign or migration will no doubt result in loss of traffic and rankings. The right steps to take are: Audit the old website in regards to content, structure, performance. It would be best if you found out what works well, which pages to keep, how much material is required, and in what topics, how to structure the content. Then it passes to doing the redirects (no more than a chain of 2), inserting the tracking pixels, setting goals in analytics, taking extra care not to create duplicate content issues, etc.
There should be no interruption in website operation for both visitors and search engines. Ideally, you work on the redesign/migration project offline and, when ready, switch to the new setup in less than an hour. The migration should happen in the less busy visiting hour of the day. It is best to put notification for visitors that you're changing the looks of the site and share the news with your subscribers.
The biggest problems I've seen with website migrations are 301 redirects, content missing, tracking pixels lost. 301 redirects are paramount to keeping some of the old rankings (not all link juice passes to the new site) and traffic (due to their past positions about to be changed when Google reads the new pages).
Additionally, you could better the site performance if you apply a responsive template to ease people browsing from mobile devices, and help with the rankings in Google's established mobile index. Speed is also a ranking factor so a responsive modern design with not many SEO blockers will help a lot.