The only solution to avoid your host sending limitation is to use an other sending server.
Shared plans have limitations on the send process but if you upgrade to a dedicated server, you will be able to send as many e-mails as you want...
Please remember your host limits the send process to avoid being blacklisted so if you use your own server, don't forget to not abuse on the number of e-mails you send per hour!
No need to change your host, simply go on the AcyMailing configuration page and select the SMTP sending method to send via another server.
If you're sending emails via SMTP through cpanel's Exim, make sure it doesn't add a delay for each email:
And other important thing to do is to make sure this option is disabled:
This option increases the SPAM level of your emails and adds a huge delay for each email sent.
If you do, please make sure the option "Embed attachments" is set to "No" on the AcyMailing configuration page so instead of embedding the file within the message, AcyMailing will just add a link to this file in your Newsletter.
If so, please disable this option... your visitor will still see your pictures inside your newsletter but instead of sending them within the e-mail, they will stay on your server.
AcyMailing has to connect to this SMTP server each time you deliver a message... it should not take more than 1 second per e-mail.
If you send via an external SMTP server, the communication with this external server takes time and sending 1 email every second is the standard rate. If you send from your own server, you will send much much faster...