Having banished all of the recurring female characters in Batman, New Look editor Julius Schwartz found it necessary to introduce some new ones. Oddly enough, the first woman introduced was not a romantic interest. It was Aunt Harriet, who appeared at the very end of Detective #328. Having heard of the death of Bruce Wayne's butler, Alfred, she decided that her nephew Dick Grayson needed another adult figure in his life, and she wasn't going to take no for an answer.
Busybody aunts are a stock character in fiction; consider the works of P.G. Wodehouse, for example. Or the Hardy Boys, whose Aunt Gertrude was a frequent foil for the youthful detectives. By the way, Aunt Harriet in the comics was nothing like the character on the TV show; she was a pretty sharp cookie and a definite threat to discover the secret activities of her nephew and his ward (as she did, briefly, before being convinced that Bruce and Batman were just friends).
In Batman #165, we first met Patrolwoman Patricia Powell. In yet another example of
DC's strong female leads in non-traditional occupations, she was the star graduate of the police academy when Batman gives out the awards one year. She consults him for a little advice on her romantic interest:
She goes on to explain that every time she's encountered Bruce, she's been wearing a mask for one reason or another. One time it was Halloween and another they both happened to be scuba diving in the same area.
In that issue and the next, we are given strong hints that a romance between Pat and Bruce is in the cards, but something always interferes with their meeting. At the end of the second story in Batman #166, there's this panel:
But that was actually it. For whatever reason, the storyline was never developed beyond that; my first guess is that France Herron, who wrote those stories and whose last Batman work came in Batman #169, was the only writer interested in the concept.
In January of 1966, as you may have heard, the Batman TV show debuted on the ABC network. One of the early episodes featured Julie Newmar as the Catwoman. She proved a popular villainess, with the same love/hate relationship with Batman that had been seen in the Golden Age.
And yet, despite that, and despite featuring her prominently in several reprints, Schwartz still did not reintroduce her into the New Look. Instead, in Batman #181, we got an obvious knockoff:
Poison Ivy has a goal to become the World's Public Enemy #1, knocking off the three current contenders, all of whom (improbably) happen to be gorgeous gals themselves. As a subsidiary goal:
The story apparently ends with Poison Ivy and her three rivals behind bars. But she warns Batman:
In the next regular issue (#182 was a reprint giant), Bruce finds himself unable to concentrate on work or crime-fighting as he moons over Poison Ivy. Meanwhile, she pretends to be on her deathbed due to her lovesickness over him. But when he visits her in the prison hospital, she tells him that she was just faking it and that she's got explosives hidden in a clump of hair:
She keeps him as a pet, but two can play that deathbed game:
Note in particular the pet jaguar; that clinches that Poison Ivy is just a knockoff for the Catwoman. At any rate, Batman puts her back in prison. That was it for her as a character in the Silver Age, although she reemerged as a much more deadly enemy of Batman later.
Catwoman herself returned a little later, in Batman #197:
Perceiving Batgirl as a rival for his affections, she decides to go into crime-fighting with the Dark Knight. But her idea of courtship apparently consists of blackmail:
He answers no, and she goes back to her life of crime. The trio of crimebusters escape eventually and capture her. In her next appearance (Batman #210), there is no hint of any romance between her and Batman.
In Detective #380, a young lady rings the doorbell at Wayne Mansion, and introduces herself to Dick and Alfred:
She turns out to be Ginny Jenkins, who several years earlier had been picked by the editors of her high school yearbook as the "girl most likely to marry Batman". But she hadn't really married Bruce; it turns out to be some convoluted bit about how she pretended to be married to him so her brother wouldn't murder him. Incidentally, that issue also featured the last Silver Age appearance of Aunt Harriet.
Ginny returned a year later in Detective #391. At the end of that story, it appeared that she was falling in love with a masseuse at Bruce Wayne's athletic club.
And that's basically it.