IMHO you pretty much have only 4 main options...
1 - Contact the publisher first and ask to submit material
The key is to balance being "persistent" with "being a pain in the ass" and burning bridges
2 - Obtain a Music Industry "lead sheet" or "pitch sheet"
There are companies that you can join that send these out to "anyone that pays" - some have a basic screening service on some songs to ensure your songs are up to the required standard - but the cynic in me says that the bar would be VERY low (or even non-existent) as there is no link back to the company that sent you the lead sheet so they don't care how good or bad you look. Some listings are shown as accepting "un-solicited" material but there are often questions raised about whether the listings are genuine so buyer beware
3 - Use a pitching service
There are lots of these available and even though they are relative expensive they are the market leader / best is TAXI. This company gives a list of what opportunities they have and they use their own "industry expert" screeners to screen songs BEFORE they are forwarded to "the industry". I can guarantee you that if you joined you will become disappointed/annoyed that your songs do not get past the screeners - and therefore never get heard by the person looking for the songs
This will be because they are looking for a very specific song, and the competition is fierce, so just on the numbers you are likely to get a lot of rejections
It depends whether you will channel this inevitable frustration in to improving your songs OR whether you will get frustrated and give up!
4 - (and the best way) - Network, Network, Network
By far the best way to get past the un-solicited barrier is to become "solicited" - and you achieve this by networking. There is no one way to do this (it ranges from moving to Nashville, London, LA etc. to getting a low paid job at a publishers. to just stalking publishers!) but it is VERY hard work
I know I keep saying this BUT the absolute FIRST thing you need to get ANYWHERE are commercially viable songs - which are VERY different from "good songs" (no matter who tells you otherwise - and regardless of the "exceptions" to this rule that can be quoted - there is a different approach to writing commercially viable songs for other artists that needs to be learnt IMHO)