As shown in your graphic, no protocol that exists "between" SMTP and POP3. SMTP is responsible for moving the message from the sending computer to the recipient's mail server. If there are multiple servers involved in moving the message from the sender's server to the recipient's server, each one moves the message via the SMTP protocol. These in-between servers (if any) are "relay" servers.
Here's a step-by-step (mail server software independent!) overview of how mail moves from sender to recipient:
- John sends an e-mail to Dave@MyMail.com
- John's computer uses SMTP to transmit to the message to his ISP's server, ISP-SERVER
- ISP-SERVER knows it isn't "authoritative" for the MyMail.com e-mail domain, so it performs a DNS lookup for one that is (specifically ISP-SERVER looks for an MX record in the mymail.com domain)
- A DNS MX record reports that the server MAIL-SERVER accepts messages for the MyMail.com e-mail domain, so ISP-SERVER sends John's message via the SMTP protocol to MAIL-SERVER
- MAIL-SERVER has a mailbox for Dave, so it puts John's e-mail in it
- When Dave checks his e-mail, his computer uses the POP3 protocol to connect to MAIL-SERVER and download the message from his friend Dave.
Most often messages are delivered like this, from the sender's mail server directly to the recipient's. However, it's perfectly possible that there could be multiple servers between the sender's and the recipient's, in which case each one acts as a "relay server" (reasons for this include having a server perform spam/virus filtering before sending the message to the destination server, or the destination company having many internal servers involved in moving the e-mail from one Internet-connected mail server to a server in the exact office where Dave works).
In any case, each relay server will look up a server that it should send the e-mail to (based on the domain name in the To: address of the e-mail) and use the SMTP protocol to pass the message on. Only once the message arrives at the destination server (where the recipient's mailbox exists) will the POP3 protocol be used by the recipient's computer to retrieve the message out of the user's mailbox.