Can I advocate for better parental leave when I'm new at the company?

I've been at my new job for 4 weeks, and I started halfway through my pregnancy. I did this knowing that my maternity leave would be 12 weeks, and mostly unpaid, but the small salary raise and better health insurance made the move a financial wash from my old job. All that said, my new employer does not have a clear family leave policy of any kind. They adhere to FMLA, and there is some short term disability, but even on their benefits website, there's no section for family leave policies. Everything is worked out individually with HR. I'm deeply concerned about the general lack of family leave in the U.S. However, I'm also new at my job, and fairly low on the power ladder (I'm a coordinator). Is there any way for me to advocate for better leave policies this early in a job? Has anyone tried to change the policies at their own companies? Successes? Failures. I'd love to hear all of your stories. Thanks!