Hi Andrew,
You can get a Search plugin and hook it on the Header part of the site. Use the same hook you’ve used for the Quote button but set a higher priority value to it so it goes after the Quote button.
Once added, the alignment can be done with CSS. We can help you with the write-up if you need help with it.
Note: As for the specific plugin, we’re not exactly sure what fits your preference but you can try hooking in search bars from plugins like Smart WooCommerce Search, YITH WOOCOMMERCE AJAX SEARCH, Advanced Woo Search β WordPress plugin, or the official WooCommerce Product Search.