Which courier service are they using? If the courier service provides same day delivery, you can find different apps that may support this courier service. Also, if no apps are available, it would be possible to integrate the courier with any of the apps if you can get the credentials and documentation for the courier.
Another option is using the FedEx SameDay City shipping. It supports almost 34 cities of the US with the sameday delivery option. You have to create an account with FedEx SameDay City and use the Shopify FedEx app to handle the shipping process.