I wouldn't do an HP without a year of history. Make sure you have your income entered on the Chase website which can help with auto CLIs opportunities without an HP.
Welcome! Do you have a banking relationship with Chase other than the Freedom Unlimited card? That sometimes helps win Chase's approval for both cards and CLIs.
It will take some time but i would conserve the HP to Chase for later after you add a bit more history.
In the meantime since you can't help but sweat the low Chase limit right now, you could add another card after researching this forum for your best chance at a higher limit start line card then that.
You just have to SEARCH and fish them out. They're there.
Really hate to see you waste a HP that can be better spent with another Lender. While waiting on some history with Chase to get some time on it, check into other options which in turn can also improve your Chase standing too.
Hope that helps but get more opinions, search the forums to compare your profile with others like it (and like FICO Scores) and set out your best possible chances.
All the best
Just to play the other side... I took a HP at 8 months to try for a CLI. I also had a $500 SL and was paying the card off multiple times a month. I requested a CLI to $5k and when I was asked why, I told them to look at my payment history and explained I wanted to use the card more. The CLI was approved