Kate and bf, musician Danny Fujikawa, took their newborn out Saturday night, on their way to dinner. It's unclear if the kid actually went in the restaurant or if she was deposited back home before the meal.

She gave birth to Rani Rose on October 2. She and Danny named the baby after his grandfather.

Hudson has 2 other kids from previous relationships -- Ryder Russell Robinson, the son of Black Crowes frontman Chris Robinson, and Bingham Hawn Bellamy, whose dad is Muse frontman, Matt Bellamy.