At first I was a little puzzled by your description of the problem wondering why an accelerator would want to intercept email. I think most of them just concentrate on HTTP traffic.
It seems it is an accelerator program that is used by more than one ISP and rebranded to suit the ISP (another ISP that uses it is JoiExpress). Apparently it is designed to intercept and speed up your e-mail by degrading (and thus compressing) any picture attachments in your email. It appears to give you no option to turn off this feature.
Anyway, I just took a look at Access4free Express.
On the HTTP front it sets up a proxy configuration in Internet Explorer using port 8080.
On the email side it seems that it probably works in a way very similar to the Avast Internet mail shield by setting up a redirection of port 110. It redirects the POP3 traffic to localhost port 8880 and so into the proxy. It then demands the login information for your ISP (and saves it so it doesn't need to ask next time) and I suspect must have a "partner" program running in the ISP that will retrieve your mail, compress it according to your compression level and return it back to the proxy which finally relays it back to your email client. In this process Avast never gets a look in.
For any Avast team or other knowlegable folks - is there a way that the mail client could use another port, have it intercepted by Avast and then have the Internet Mail provider still direct the POP3 requests to port 110?
Alternatively, it may be worth a test of tricking the accelerator. It may work if, in the Internet mail provider Redirect tab, you set the POP3 port to 8880. Leave the port in the mail client set to 110. You will need the "Ignore local communication" box unchecked.
Then it should go mail client > 110 > localhost:8880 > Avast > Access4free Express and back through the same route.
Try it out.