Can you haggle car insurance quotes down when it’s not a renewal?

I’m switching cars and trying to get my insurance down, i’ve used all the tricks like job picker and adding my parents on, but money is tight and im still hoping to reduce it further. I’m wondering if i can call the company with the lowest quote and see if they’d lower it any further for me. I’m just not sure if i even have any wiggle room with this as it’s not a renewal quote and it’s the lowest quote. Any tips or advice?