This is an interesting question, so I'll take a stab at it. Direct evidence for the claim is hard to come by. Generally, religious affiliations, conversions, and loss of faith are self-identified in surveys. The reliance on self-identification makes it difficult to test the claim that people move to some other non-evidence-based reasoning, as they may not be aware of it. Indirect evidence is easier to find.
The extensive Religious Landscape Survey (2009) by the Pew Forum on Religion & Public Life for example, asks the self-identified "unaffiliated" who grew up religious to explain their reasons for leaving their faith. Only about 40% leaving Christianity cite not believing in God as their reason, and less than 30% leaving Catholicism cite no longer being religious in some way. Looking at specific survey responses, most converts leave their religion for reasons other than relying on science and logic, and remain "spiritual" in some way.
Another way to test the claim is to ask people about their beliefs, as is nicely summarized here, from the same survey. A large proportion of non-religiously affiliated participants - even some self-identified atheists - believe in God, the afterlife, or some other type of spirituality.