I recently wandered down two very useful timelines. They’re both on the Cold Spot Paranormal Research website, and the first timeline looks at ghost hunting in the 1800s. The second continues this into the 1900s. Michelle McKay, who compiled these histories, was very smart to include some useful context. For instance, she marks key historical periods, such as the Victorian era, and notes important dates in related movements, such as Spiritualism.
Along the way, McKay raises the question of when the term “ghost-hunter” appeared. This is something I’ve wondered about, too. The Oxford English Dictionary — at least, the free version online — traces the phrase to 1894, when Andrew Lang used it in Cock Lane and Common Sense. To be sure, Lang even has an entire chapter there titled “Presbyterian Ghost Hunters,” suggesting the term was pretty well established by then.