She's very sweet.
I don't think babies can really 'look' like any particular name but here are some 'unusual but not too out there' suggestions that have a similar vibe to June in my opinion. (I wouldn't actually use June if all you family and friends hate it. It was very rude of them to say so but I'd be really put off by knowing that everyone who interacted with my baby hated her name
)
Averil, April, Ariel, Astrid, Bronte, Bryony, Carys, Celest, Cecily, Coralie, Daphne, Della, Edith, Effie, Esme, Faith, Greta, Hazel, Iona, Ida, Inez, Judy, Jocelyn, Joy, Julie, Lara, Lois, Margot, Mara, Nina, Nadine, Polly, Penny, Rowena, Renee, Sadie, Sally, Sarah, Tara, Tessa, Tamara, Una, Verity, Willa, Yvette