You won't be able to have that done using the standard Constant Contact sign up process. As you noted the best you can do is the next day.
If this is a real business process requirement your going to have to design your capture system using something else to notify you right away and then enter the customers info that it captures into CC as a separate process.
We have done this for a few clients. We design a web form based on your requirements that would go on your site, that form when filled out would send you an instant notification via email and STORE the customer info into an Excel format. We then import that on your behalf on a weekly basis. The import frequency can be increased or decreased based on your needs.
This also allows you to keep your customer on your site and also redirect the sign up to any page you want after he completes the form.
Pretty cool stuff, that Constant Contact does not currently offer.
If this is something that interests you, we'd be glad to give you a quote on this custom programming and import services.
You can see an example on our website below on the Contact Us page.
Accredited CTCT GOLD Solution Provider
Constant Contact All Star 2009-2010-2011-2012