To be honest, I've never really gone chasing after Laurence D. Smith, but I suspect this is the same fellow:
www.philsp.com/homeville/cfi/z38.htm#A3639Smith, Laurence Dwight (1895-1952) (stories)
Born in New York City; graduated from Yale and went into the real estate business, then into writing novels and true crime books; died in the Veterans Hospital in Albany, New York.
If Laurence D. Smith and Laurence Dwight Smith are indeed the same writer (which seems highly probable), then he published a smattering of stories in the pulps around the mid 1930s. Along with his work in THE WITCH'S TALES, the Cook-Miller index says he appeared in THE UNDERWORLD MAGAZINE and TEN DETECTIVE ACES circa 1934 - 35. The most intriguing title I spot among those is "The Ghoul of Longwood Cemetery." But despite the creepy title, I suspect the yarn is more mystery/detective than horror.
From roughly 1936 to 1941, Laurence Dwight Smith published hardcover mysteries with houses such as Lippincott and Mystery House (according to Alan J. Hubin's index). He also had a short story, "Seesaw," in a 1942 issue of ELLERY QUEEN'S MYSTERY MAGAZINE. A quick Google search also indicates he wrote books on true crime and cryptography. Seems to have mostly gotten out of the fiction market in the early 1940s.
I have no clue how he got tied up with THE WITCH'S TALES, nor how he wound up adapting "Mrs. Hawker's Will." He was almost a ghost writer on that one, as the magazine declares in large type that the story is by Alonzo Deen Cole, and then practically hides "adapted by Laurence D. Smith" in much smaller print on a completely different portion of the page. I do notice that both THE WITCH'S TALES and THE UNDERWORLD MAGAZINE were from Carwood Publishing, so I'd guess that LDS got the gig writing for WITCH'S TALES through his acquaintance with one of Carwood's editors. Interesting that issue #2 of WITCH'S TALES would hire someone to adapt Cole's story, when all of the other WT pulp magazine entries purport to be from the pen of Alonzo Deen Cole himself.
Thanks, BTW, for your interest in my proposed WITCH'S TALES collection. I did a lot of work on it back in 2012, but a setback has left it on the back-burner since then. Corresponding with Helrunar is what inspired me to compile the stories in the first place, and he owns one of only four dummy editions of the volume I produced. Hopefully, I can get that project back on track sometime soon. Or at the very least, I can post the introduction I wrote for it here on the forum.