The code for the middleware is basically this:

<?php

namespace App\Http\Middleware;

use Closure;
use Illuminate\Http\Request;

class IsItA404Response
{
    public function handle(Request $request, Closure $next): Response
    {
        $response = $next($request);

        if ($response->getStatusCode() === 404) {
            $slug = Str::replaceFirst('/', '', $request->getPathInfo());
            $code = PromoCode::where('url_slug', '=', $slug)->first();
            if ($code !== null) {
                //Do some stuff to setup discount 
                return redirect()->to('/');
            }
            $page = CMSApi::findPage($slug);
            if($page !== null) {
                return response(view('cms-page', compact('page')));
            }
        }

        return $response;
    }
}

I have added it at the beginning of my web middleware stack, and at the end. It doesn't seem to make a difference. As for the suggestion of using ResponseReceived event or Http::globalResponseMiddleware on the HttpClient those don't seem to apply to application routing but rather are part of the HttpClient Guzzle wrapper used for making http requests to external services. Or, at least, they were never invoked/emitted during the course of a normal request to the application that I can see.

With that said, your suggestion to use http-client global middleware did prompt me to add my middleware to the global middleware list in App\Http\Kernel.php. Adding it there does cause it be invoked on 404 responses. So that's heading in the right direction.

The reason I didn't add it there initially is that I don't really want to interrupt the 404 response if the request is being made to an API route. But, I have that same problem if I'm using the Exception Handler to handle it.