Middle names for first name Kate?

Expecting a baby girl in a couple of months and are stumped on her name! We love the name Kate (not Katherine, not Katie, just Kate), but the problem we have is that every middle name we try to place with it seems to sound better flipped around, with Kate as the middle name instead. Just a few examples of this:

-Kate Hannah (we think is sooo darling as Hannah Kate instead)

-Kate Ellie (Ellie Kate sounds more natural I think)

-Kate Amelia (Amelia Kate sounds better to me!)

Is it just us?! Or is Kate a really tough name to find a middle name for?!

We are partial to J middle names, as our other kids have J middle names, but can't settle on one we like, especially not with Kate. I think the only J name we both like and haven't used is June, but Kate June sounds a little choppy maybe? Grace is another middle name we would like, but again maybe choppy with the single syllables.

Again, this is totally maybe just our preference, I don't mean to hurt anyone's feelings about their names they have chosen! Any Kate moms who have middle name suggestions? Help please! Thank you in advance!