There are a couple of useful plugins when running a static website exported with Simply Static.
Autoptimize is a free plugin that optimizes your assets (CSS/JS) and can also optimize your HTML output. It’s pretty easy to set up and provides good results in optimizing the rendered output of your website.
It can greatly improve the speed of your static site as well.
Contact Form 7
When using a form on your static website I would recommend using Contact Form 7. It is easy, free, and pretty flexible. Also, it’s directly integrated with Simply Static to use on a static website.
WP Offload Media
Depending on your static hosting setup you need to decrease the overall size of your website by loading your images via another CDN. The best plugin for the job is WP Offload Media.
It allows you to sync your media files to an Amazon AWS S3, Digital Ocean Spaces, or Google Cloud Storage storage. This way you can easily use for example Cloudflare Pages to host larger websites.
Better Search and Replace
After you have created a static version of your website, chances are good that you want to change the domain of your old WordPress website for administration. For example from admin.patrickposner.dev to manage.admin.patrickposner.dev while the static websites is routed to admin.patrickposner.dev. The Plugin Better Search and Replace is a really good tool to replace all URL’s within your website.