How to navigate raising rates after accidentally lowballing yourself?

My friend (a Nigerian, 25F) who has been out of work for some while recently was recommended for a role at the company of one her ex directors to come aboard as a freelancer/contractor at their new company.

Luckily, the ex director, now CEO of the new company (based in the US) already knows about her work to some extent so there wasn't much screening.

They arranged a meeting after she sent over some of recent works and during the meeting she quoted her hourly rate as $20 but the CEO was complaining that they're still starting small and $20 seems much and since she'll be getting work and the role might eventually develop into a more proper FT engagement kind-of, that they can't afford $20 right now.

So, my friend out of enthusiasm (for the long term she heard) blurted $12 and they agreed on that.

After the meeting, she's complaining that she had undersold herself and should've negotiated for $15 lowest.

I already advised her to wait until 6 months (hopefully, she still has the job), that if her role develops into something more serious then or not, she can bring it up that she wants to raise her rates. I even gave her some pointer on what to say

"Now that I've settled properly into this role, I want to bring to your notice that I would like to raise my rates to an amount that can ensure I continue to deliver quality work for the organization"

She's concerned that 6 months is too far but at the same time doesn't want to be too quick as she feels she should also gain the trust of this new organization.

I've not really worked in many organisations myself so I feel like I might not be in the best position to give a good advice.

So, I'm looking for other perspectives on what I can tell her on how to navigate this.