The phrase "under hat" is a rather unique one, and doesn't really have any synonyms that convey the same meaning. Typically, when we think of something being "under hat," we might be referring to something that is hidden or kept secret. In this case, we might use words like concealed, undisclosed, or confidential. Alternatively, if we are using "under hat" to refer to someone who is responsible for or in charge of something, we might use phrases like in control, holding the reins, or at the helm. Regardless of the specific phrase we use, the underlying idea is that something important is being kept safe and secure.