Anthony Gilbert is the pseudonym of a woman named Lucy Malleson whose Mr Crook mysteries were published between 1936 and 1974, which is a pretty good run. Unfortunately all of her books appear to be out of print.
Which, judging from The Spinster's Secret, is a shame. Published in 1946, the story is about an old lady, Miss Martin, whose callous niece moves her out of her little one-room flat into a home for old ladies. When she realizes a little girl she had known in her happier life is now one of dozens of listless girls in a nearby orphanage she answers an ad in the paper for a private detective.
Mr Crook, who is dumpy, has an unfortunate accent, and wears loud plaids, is the first person who believes her suspicions and who takes seriously her worries about the little girl. The well-meaning lady who runs the home where Miss Martin lives stops her from making a crucial phone call and the results are disastrous.
I figured out what was going to happen before it occurred and could foresee what was coming, but the plot is not the heart of this wonderful little story. When I began reading it I thought the author was a man and I was astonished at his understanding of and empathy with an ageing lady who must rely on a relative who abuses her power. I was not surprised to learn that Lucy Maleson was a lonely spinster herself. I was reminded of Elizabeth Taylor's Mrs Palfrey at the Claremont (which was published in 1971).
This title didn't appear to me out of thin air, but it may as well have as I can't figure out who posted about it in a blog. If you were that blogger, please let me know so I can thank you properly. This is a treasure.
Update: Col at Col's Criminal Library blog, which is worth a visit, by the way, reminds me that Moira at Clothes in Books reviewed this book recently. And that is, indeed, where I learned about it. I'm most grateful as this is a mystery with the most carefully developed characters I've encountered in a while. Do visit Clothes in Books and read the reviews Moira posted here and here.