I love [name]Julia[/name] [name]Catherine[/name].
I love [name]Julia[/name] [name]Claire[/name] more than [name]Julia[/name] [name]Clare[/name] because the spelling [name]Claire[/name] helps [name]Julia[/name] to kick up her heels. [name]Julia[/name] [name]Clare[/name] feels a lot more serious. But it is very lovely, too.
Hmm…I’m not sure if they’ve been suggested before, but you may also like:
[name]Amelia[/name]
[name]Alice[/name]
[name]Annabelle[/name]
[name]Clementine[/name]
[name]Celia[/name]
[name]Charlotte[/name] (although it has the same root as your name)
[name]Cora[/name]
[name]Diana[/name]
[name]Eleanor[/name]
[name]Eliza[/name]
[name]Emmeline[/name]
[name]Frances[/name]
[name]Felicity[/name]
[name]Flora[/name]
[name]Genevieve[/name]
[name]Hope[/name]
[name]Helena[/name]
[name]Ivy[/name]
[name]Juliet[/name]
[name]Katherine[/name]
[name]Louisa[/name]
[name]Lydia[/name]
[name]Miranda[/name]
[name]Margaret[/name] (nn [name]Maisie[/name] or [name]Daisy[/name])
[name]Penelope[/name]
[name]Phoebe[/name]
[name]Rose[/name]
[name]Rosalie[/name]
[name]Ruby[/name]
[name]Scarlett[/name]
[name]Sarah[/name]/[name]Sally[/name]
[name]Tess[/name]
[name]Victoria[/name] (reminds me of [name]Alexandra[/name])
[name]Vivian[/name]/[name]Vivienne[/name]
[name]Willa[/name]
Good luck, [name]Caroline[/name]! 
[name]Julia[/name] [name]Claire[/name] is very pretty I also so the suggestion for [name]Julia[/name] [name]Sinclair[/name] and think that’s pretty too.
Some one suggested [name]Julia[/name] [name]Clio[/name] or [name]Julia[/name] [name]Cleo[/name] and I think that sounds really sweet.
[name]Julia[/name] [name]Natalie[/name] [name]Claire[/name] is very pertty 
As far as other pretty classic names - [name]Jill[/name] gave great suggestions it will be hard to add more. Some of my suggestions are not so much classic as they are pretty.
[name]Anastasia[/name]
[name]Annmarie[/name]
[name]Christina[/name]
[name]Dahlia[/name]
[name]Julianne[/name] - a bit more modern than classic
[name]Lauren[/name]
[name]Lydia[/name] - [name]Jill[/name]'s suggestion but it’s my personal favorite that I can’t use.
[name]Matilda[/name]
[name]Miriam[/name]
[name]Marian[/name]
[name]Natalie[/name] or [name]Natalia[/name]
[name]Paulina[/name]
[name]Rosemarie[/name]
[name]Seraphina[/name]
[name]Thalia[/name]
that’s a good one! i thought of using it but i chose johanna, & charlotte is my middle name so i chose julia caroline:D