In case you need to access a specific set of remote services you could use an intermediate reverse-proxy, to perform tls1.2 for you. This would save you from trying to patch or upgrade java1.6.
e.g. app -> proxy:http(5500)[tls-1.2] -> remote:https(443)
Configuration in its simplest form (one port per service) for apache httpd is:
Listen 127.0.0.1:5000
<VirtualHost *:5500>
SSLProxyEngine On
ProxyPass / https://remote-domain/
ProxyPassReverse / https://remote-domain/
</VirtualHost>
Then instead of accessing https://remote-domain/ you access http://localhost:5500/