OK, before I conclude that this writting is bad, maybe someone here can help me realize I missed something...
We see Spica setting her sights on the idol Yamaguchi Neko (sorry if I misremember the name, but it was something like that). She gests a part-time job to earn money so she can go to the debut concert and participate in the hand-shake event. She does it and when shaking hands she reads the idol's heart and discovers OR confirms her past as a murderer.
When confronting the idol, she made it sound like it was a coincide that her beloved idol turned out to be a murderer. And not only that, she just happens to be the murderer of the case this old detective, who happens to be a client of her part-time job, couldn't solve.
If the author of the manga really want to pass all this as pure coincidence, then this writting completely sucks. If there was some clue I missed that could potencially link the idol to the unsolved case, then this could be salvaged. i guess it all comes down to if Spica just discovered the murder in the handshake event, or she knew about it before hand.