In case you face some issues with the popular email providers, please check whether you have allowed the usage of the third-party apps. The problem may be in the security limitation and without changing some security settings you may face the incorrect work of the SMTP extension.
The solution for Gmail can be found here. You may find the desired manual for your other email providers in their help sections or seek help via their support teams.
Try to reach the mail server via the console (CLI).
For example, try to ping or use telnet smtp.hostname.com:port directly from your server where Magento is hosted (not PC) and check the response in the command line. If it fails to reach the mail server, the connection is blocked and you'll see 'connection timed out' error message.
Alternatively, you may contact your hosting provider to make sure that outbound connection to the specified port is not blocked from their side.
The reported issues usually occur if any other SMTP or third-party extension is installed in your instance.
It may provoke a conflict which causes rewriting classes responsible for email sending and logging.
Please, first of all, check whether the Amasty SMTP module has conflicts with other extensions by going to System > Configuration > Amasty Extensions > Extensions and Notifications > Troubleshooter > Show:: open Models section.
If there are any, you will see the notification in red. In order to check whether other modules affect Amasty SMTP extension functionality, please disable them temporarily via .xml configuration file.
For that, follow the guide below:
Edit the following configuration file: app/etc/modules/Extension_name_X.xml (where X is the name or the module you want to disable).
Change "true" to "false"
Clear Magento cache
Once done, the extension will not affect your Magento anyhow.
The list of 3rd party extensions which usually conflict with Amasty SMTP extension:
Fooman Email Attachments
Arschorder SMTP Pro
The second advise to check is Magento cron.
Our SMTP extension relies on cron performance. In order to send order confirmation emails & invoicesit is required to set up cron.
Here you can learn more of about Magento cron and how it should be configured.