You have to write dynamic jQuery code to achieve this and no need to use any hook.

Awesome great to see you got that sorted.

Please advise if you have more questions.

Have a fantastic day!

I want the sliding text bar to fill up the entire space available in the container

Then please use the Full Width style in the plugin Menu Search settings instead of the Sliding style.

I hope you are well today and thank you for your questions.

Is there a special hook I can use with this plugin so that it will refresh the page after adding an item to the cart?

You have to develop custom javascript code to achieve this.

Developing custom code for custom functionality is beyond the scope of support that we provide here.

If you are not a developer then you can consider hiring a developer to develop it for you. You can hire a developer from any freelance site or WordPress agency.

You can also consider us to develop it for you using the form https://ivorysearch.com/custom-work/

Best regards,
Ivory Search Support

You are always welcome, I’m glad I was able to help you with this matter. There is no obligation but by your kind words I wonder if I could trouble you to leave a review based on this experience here:

https://wordpress.org/support/plugin/add-search-to-menu/reviews/?filter=5

I’d really appreciate that. 🙂

And if I can be of any further assistance please don’t hesitate to ask.