Searching for Private Schools

I do well at my job, I take it seriously, and I hope to get the sponsored license in the future when I've improved my language abilities (currently N3).

I know most private HS roles are through contacts/connections/word of mouth but I dont really have any contacts in those opportunities. So, im thinking of emailing private high schools in my surrounding area and a bit further out.

Is this the best way to go about it in my situation? Finding them first is a little mission, cold-emailing them is a little awkward etc. But I'm willing to do it if its possible. Im also keeping an eye on GaijinPot.

I just want to move up a little from where I am now, monthly salary-wise. SHS ALT is preferred.

cheers