I rather doubt you see any appreciable difference in speed with either http or https.
Whilst https is secure encrypted, so the traffic must be encrypted and decrypted, this has an overhead, but that overhead happens on the local system or server before or after sending/receipt once encrypted there really shouldn't be any difference in transmission speed.
The problem with using https is that avast's web shield can't monitor that traffic so you lose a bit of security. That said you 'may' gain a little as the web shield isn't scanning it, but that again would barely be noticeable due to being on dial-up (the web shield isn't going to be snowed under by slow dial-up).