Is it possible to send a reminder to unconfirmed contacts so they subscribe?

Yes you can.

Set up an autoresponder for your contact list.

When you are creating it, make sure you set the first page (with all of the filter options) to send to UNCONFIRMED contacts.

Then inside the autoresponder content, include '%%CONFIRMLINK%%' somewhere and the application will pick that up and change it into a link they can click to confirm their details.