Reason is a broader concept that includes different forms of thinking, including thinking logically and draw conclusions based on evidence, experience, and principles. It involves the broader process of thinking and understanding, including both logical and intuitive elements, while logic is more specific and deals with the formal rules of inference and deduction.
So, if you’re looking to use reason without the logical aspect, then I think that would be a focus on the aspect of using intuition without the support of logic.
Immanuel Kant's wrote "Critique of Pure Reason" (1781) where he distinguishes between sensible intuition and intellectual intuition. Sensible intuition is related to empirical perception, while intellectual intuition is a direct intellectual understanding without needing sensory experience. However, Kant believed that we needed the use of concepts and logic in order to make sense of intuitions.
Henri Bergson wrote "An Introduction to Metaphysics" (1903) in which he argues that logic and analysis fails to capture the basic nature of reality, which can only be understood and interpreted through intuition.
Georg Wilhelm Friedrich Hegel, who wrote “Phenomenology of Spirit” (1807) in which he suggests that intuition provides us with immediate awareness of reality, while logic, which he calls dialectics, is important for understanding the rational structure that builds our reality.
I don’t know if there’s an argument that intuition can be linked to instinct, but it often is expressed as an insight into something without the need for conscious analysis or explicit reasoning. Another way of thinking of it is that intuition is a form of knowledge or understanding that is gained without conscious reasoning. Considering that unconscious form of reasoning, does it count toward the OP’s question?
And, if so, then I’d have to question whether or not the process of insight and the use of implicit knowledge subconsciously could be considered using reason without logic, or if it is instead a system of rapid processing of logic that bypasses the slower conscious method.