Updating Image Links from http to https

When I switch my WordPress site from http to https in the settings, WordPress continues to call images over http links. Is there an easy fix so that images are called over https?

I wouldn’t care except that my site has static pages in straight html, and I only use WordPress in a separate directory under the same domain name. My order forms are on those static pages. When I put the Apache mod in my htaccess file to force all traffic to the https address, it works great in the static pages, but in my blog, all the image links get broken. This is because the image links are still looking for http, and after the Apache mod, there is no http.

I tried the Velvet plugin, but it didn’t change the image links. I’m nervous about another popular plugin that hasn’t been updated in two years. I also found some instructions for running a search and replace in the SQL database, but that’s a process with which I have zero familiarity. Any recommendations on the easiest, most foolproof fix today?