If you're looking for manga similar to Mr. Bojangles, you might like these titles. All recommendations are made by Anime-Planet users like you!
Kanae's husband disappeared without a trace while on a union trip. Though usually headstrong and independent, the woman becomes plagued with worry over what happened to him, and is unable to move on with her life. Thus, Kanae decides to hire a private investigator, and keeps running the public bathhouse to the best of her abilities, but the detective who shows up is disheveled and eccentric. As he tries to figure out the truth about her husband, Kanae must deal with the stress from a work-intensive job, meddlesome neighbors, and recurring nightmares in which she is drowning.
Both of these manga by Tetsuya Toyoda have a private detective hired to find a missing person. In fact, I'm pretty sure it's the same detective, although he's only named in Undercurrent.
If you didn't like Mr. Bojangles that much, don't let it discourage you about Undercurrent, as there's a lot more going on than just the detective.