How do you get inspiration for product design when it comes to designing a new product?

It sounds like you're potentially looping your designer in too late. They should be there early to help you think through competitors, potential inspiration in the market, and help you learn from users. Getting a user's first explanation of their painpoints is one thing, but you will develop your understanding of their painpoints by showing them potential solutions (that you and your designer work on together).

That all said, if I'm short of immediate inspiration, I like to look laterally, outside my immediate product space, for inspiration. For example, when I was working on how to make individual work rewarding in Asana, I looked at how fitness apps help users feel a sense of reward and achievement. If something works in another use case, there's likely something you can learn from and adapt to your use case.