my website not responding with SSL - apache

Question

I have a website hosted on a VPS running CentOS 6 and running fine. Now I have to make it accesible with SSL, so I configured /etc/httpd/conf/httpd.confand /etc/httpd/conf.d/ssl.conf to make httpd listen to port 443, then binding the virtualhost related to the website to the same port.
The apparent result of the whole configuration may be resumed by the following diagnostics that also resumes some tests I performed (I masked data where necessary, and my.site.com is dummy):
# service httpd restart
Stopping httpd: [ OK ]
Starting httpd: [ OK ]
# httpd -S
VirtualHost configuration:
wildcard NameVirtualHosts and _default_ servers:
*:443 my.site.com (/etc/httpd/conf.d/ssl.conf:74)
*:80 my.site.com (/etc/httpd/conf/httpd.conf:1022)
*:8888 my.site.com (/etc/httpd/conf/httpd.conf:1045)
Syntax OK
# openssl s_client -connect localhost:443
CONNECTED(00000003)
depth=2 O = [Some authority here], CN = [Some authority here] Root CA X3
verify return:1
depth=1 C = US, O = [Some authority here], CN = [Some authority here] X3
verify return:1
depth=0 CN = my.site.com
verify return:1
---
Certificate chain
0 s:/CN=my.site.com
i:/C=US/O=[some authority here]/CN=[some authority here] X3
1 s:/C=US/O=[some authority here]/CN=[some authority here] X3
i:/O=[some authority here]/CN=[Some authority here] Root CA X3
---
Server certificate
-----BEGIN CERTIFICATE-----
[some certificate codes here]
-----END CERTIFICATE-----
subject=/CN=my.site.com
issuer=/C=US/O=[some authority here]/CN=[some authority here] X3
---
No client certificate CA names sent
Server Temp Key: ECDH, prime256v1, 256 bits
---
SSL handshake has read 3672 bytes and written 373 bytes
---
New, TLSv1/SSLv3, Cipher is ECDHE-RSA-AES256-GCM-SHA384
Server public key is 4096 bit
Secure Renegotiation IS supported
Compression: NONE
Expansion: NONE
SSL-Session:
Protocol : TLSv1.2
Cipher : ECDHE-RSA-AES256-GCM-SHA384
Session-ID: [some coding here]
Session-ID-ctx:
Master-Key:
[some coding here]
Key-Arg : None
Krb5 Principal: None
PSK identity: None
PSK identity hint: None
TLS session ticket lifetime hint: 300 (seconds)
TLS session ticket:
[other codes here]
Start Time: 1483709215
Timeout : 300 (sec)
Verify return code: 0 (ok)
---
closed
# openssl s_client -connect [my.site.com's IP] -state -nbio 2>&1 | grep "^SSL"
SSL_connect:before/connect initialization
SSL_connect:SSLv2/v3 write client hello A
SSL_connect:error in SSLv2/v3 read server hello A
SSL_connect:SSLv3 read server hello A
SSL_connect:SSLv3 read server certificate A
SSL_connect:SSLv3 read server key exchange A
SSL_connect:SSLv3 read server done A
SSL_connect:SSLv3 write client key exchange A
SSL_connect:SSLv3 write change cipher spec A
SSL_connect:SSLv3 write finished A
SSL_connect:SSLv3 flush data
SSL_connect:SSLv3 read server session ticket A
SSL_connect:SSLv3 read finished A
SSL handshake has read 3672 bytes and written 373 bytes
SSL-Session:
SSL3 alert read:warning:close notify
SSL3 alert write:warning:close notify
# iptables -L -v | grep https
0 0 ACCEPT tcp -- any any anywhere anywhere tcp dpt:https
# netstat -lnp | grep 443
tcp 0 0 :::443 :::* LISTEN 22682/httpd
Now, the website is still responding via the usual browser http://my.site.com address, while it return ERR_CONNECTION_TIMED_OUT if trying https://my.site.com.
Furthermore, accessing by remote I obtain:
# openssl s_client -connect [my.site.com's IP:443]
connect: No route to host
connect:errno=113
So, what is the problem or how can I investigate its origin?
Thanks in advance for your help.