I have a friend who used to work with the government and now he is working for a private company / agency via a shipping firm. He is in Qatar currently working in a shipyard and we once discussed if it is possible to have a GSIS account integrated to your new SSS account or vise versa? He promptly answered that it is possible only if you come from a government owned or managed firm and you want your remittance as a member be transfered to your new account which is with a private company in your SSS.
On the other hand, according to him, it is not possible to have your remittance transfered or integrated with your new GSIS account if you came from a working status where you used to work for a private firm. Meaning, if you want SSS account integrated with your GSIS account with respect to remittance, it is not possible. Go ask the nearest GSIS or SSS branch for your case if you have the same as my friend. In the meantime, be sure to ask what is the best option for you if you are in this situation.