One of the first things you can do is to check with the US Copyright Service and use their searchable database. In a lot of cases, the copyright may have been passed on to someone else as part of an inheritance, or it may have been sold to someone else. Using this database to search is a good first step. Keep in mind that the original author may not be the person who renewed the copyright. It may have been renewed by an heir, an estate, or a publisher, depending on whether or not the rights were transferred.
Something else you may need to be cautious about is whether or not something was later included in a collection of works which might have its own copyright. This is especially true with things like short stories, poems, cookbooks, and other smaller works. While the author or heirs may not have renewed the copyright on the individual work, they might have a copyright on that smaller work as part of a larger collection.
As an example, if you published a short story as a chapbook in 1970, it would be subject to copyright for 28 years, until 1998. However, if you took that same story and included it in a short story collection with some of your other stories in 1976, then the copyright for all stories within that collection would last until 2004. Even worse, if the short story collection was published in 1978 (which is when copyright laws changed), then each story would be protected until 70 years after the death of the author(s). (I actually experienced a situation like this with a story in which I was considering writing a derivation.)
You also need to keep in mind how you are going to go about publishing. If you are planning to use something like Kindle Direct Publishing to self-publish as an e-book, then you need to be aware of the limitations imposed by Amazon. Other e-book publishers have specific guidelines for public domain books as well, and some are even stricter. The most basic guidlines for Kindle are that the new work has to be differentiated in one of the following ways:
• (Translated) - A unique translation
• (Annotated) - Contains annotations (unique, hand-crafted additional
content including study guides, literary critiques, detailed
biographies, or detailed historical context)
• (Illustrated) - Includes 10 or more unique illustrations relevant to
Books that meet this criteria must include (Translated),
(Annotated), or (Illustrated) in the title field.
Something else to keep in mind is that certain characters are protected either by Trademark or other legal means. As an example, even though the original Sherlock Holmes stories are now considered to be in the public domain, the rights to the use of the character are still protected by the author's estate and cannot be used without their permission. This is true for countless other major literary characters as well.