A multi-file search-and-replace would be a very good idea.
The following code snippet will work as a temporary work-around, assuming that you already have other working mod_rewrite rules in place, and that you place it before all of your existing redirects (because the target domain is different), and that all of your redirects are coded using mod_rewrite and have not mixed in mod_alias "Redirect" and/or "RedirectMatch" directives. In this latter case, there is no way to predict whether the mod_rewrite or mod_alias directives will be processed first, since directive invocation is per-module, and not in strict line-by-line order.
As previously stated, the old server must be publicly- and independently-accessible either via domain name, subdomain, or IP address.
# Redirect requests for missing images to old server
RewriteCond $1 !^excluded-local-image-directory-path-1/
RewriteCond $1 !^(excluded-local-image-directory-path-1|excluded-image-directory-path-2)/
RewriteCond %{REQUEST_FILENAME} !-f
RewriteRule ^(.+\.gif)$ http://www.old-example.com/$1 [R=302,L]
I included two 'token' RewriteConds at the top. If possible, put the name(s) of the new sites image directory-paths into these RewriteConds, so that the file-exists checking can be skipped if the image can be safely presumed to exist on the new site based on its URL-path.
I show two forms, the first excluding one directory-path per RewriteCond, and the second using a "local OR" to exclude more than one local image-directory-path. You may use either or both forms, adding more RewriteConds in these formats if needed. Do not end-anchor these patterns or they won't work.
Alternately, if the directory paths for missing images are all the same or if there are only a few, you may wish to use a positive-match pattern either in RewriteConds or in the RewriteRule pattern itself, so that only images in directory-paths which are likely to be missing will be checked. The decision should be based upon which approach results in the fewest directory-path inclusion/exclusion directives and/or the simplest and smallest code. However, even excluding or including several dozen directory-paths will still likely be faster than one file-exists check, so use as many as you need to avoid unnecessary disk checking.
The rule above only redirects requests for missing .gif files. If you wish to include other types, its a simple matter of including them in the RewriteRule's pattern, e.g. "^(.+\.(gif|jpe?g|png|ico))$". However, be aware that the more image-types you have to check, the more disk checks you will invoke, and the slower this code will get; Include only the filetypes that are necessary, and put them in order, most- to least-used (check your "stats").
Using the local image directory exclusions/inclusions may make the site's loading speed tolerable, despite all the file-exists checking and client redirects for 'missing' images, but I doubt that you or your visitors will be very happy with the performance of this solution if there are lots of missing images.
Jim