It's possible to OCR the PDF after the scan but not with Reader. You would need the full version of Acrobat (or a different OCR product).
Whoops, that looks like £500.
Yeah. It's a bit spendy but it does a lot more than just OCR.
If you don't need any of the rest of the fluff, you can try Abbyy FineReader. It's a dedicated OCR application and in most peoples opinion does a better job at it than Acrobat.
Only $99 here in the states (they also offer a free trial period). It may differ depending on where you are located.