When sending mail via PHPmailer() scripts, do you get this in the logs ?

Date TLS error on connection from localhost (www.domain.com) [127.0.0.1] (SSL_accept): error:140760FC:SSL routines:SSL23_GET_CLIENT_HELLO:unknown protocol

The reason may be old PHP scripts are trying to connect via  TLSv1.0 and TLSv1.1 and it may be disabled in the server. While its best to upgrade the scripts, in case you want to enable  TLSv1.0 and TLSv1.1 here’s how you do it in exim

Open /etc/exim.conf and change the following two parameters

openssl_options = +no_sslv2 +no_sslv3

tls_require_ciphers=ECDHE-ECDSA-CHACHA20-POLY1305:ECDHE-RSA-CHACHA20-POLY1305:ECDHE-ECDSA-AES128-GCM-SHA256:ECDHE-RSA-AES128-GCM-SHA256:ECDHE-ECDSA-AES256-GCM-SHA384:ECDHE-RSA-AES256-GCM-SHA384:DHE-RSA-AES128-GCM-SHA256:DHE-RSA-AES256-SHA256:AES128-SHA:AES256-SHA:DES-CBC3-SHA:!DSS

Once done, restart the exim service.

If you are doing it via WHM, Go to Home »Service Configuration »Exim Configuration Manager :

1) Search for 'Options for OpenSSL' and pass the following value in the box

+no_sslv2 +no_sslv3

2) Search for 'SSL/TLS Cipher Suite List' and pass the following value in the box 

ECDHE-ECDSA-CHACHA20-POLY1305:ECDHE-RSA-CHACHA20-POLY1305:ECDHE-ECDSA-AES128-GCM-SHA256:ECDHE-RSA-AES128-GCM-SHA256:ECDHE-ECDSA-AES256-GCM-SHA384:ECDHE-RSA-AES256-GCM-SHA384:DHE-RSA-AES128-GCM-SHA256:DHE-RSA-AES256-SHA256:AES128-SHA:AES256-SHA:DES-CBC3-SHA:!DSS