Applying to a New Job Before Even Starting My First?

I recently graduated from a top target and will be beginning my career in Tier 2 consulting (OW/S&/LEK) this July. I am interested in lateraling to private equity when I get the chance and seeing that I am not at a BB or MBB, I am worried that PE headhunters will not be reaching out to me this winter. I want to be proactive this summer before beginning work (I know the odds are stacked against me coming from my current role) and want to know if it is advisable/ok to reach out to headhunters to ensure that I am on their radars?

I also interviewed with a few PE firms during this past school year and am considering following up with them to talk about their processes moving forward. Is it a bad look to reach out about future opportunities even before I have set foot at my new job?

And lastly, I recently came across a few job postings on my school's career database with 2019 start-dates and am interested in applying. Again, should I not be applying to these before I begin work?

Thanks for the help!