Non-classical logics can be divided into three categories. There are those that may be called sublogics, because the set of their theorems is a proper subset of the theorems of classical logic. There are extensions of classical logic, which have the property that the set of their theorems is a proper superset of those of classical logic. And there are contra-classical logics, whose theorems differ from those of classical logic.
An example of the first type is intuitionistic logic. It differs from classical logic by lacking the law of excluded middle and double negation elimination. There are also substructural logics that lack structural rules such as contraction or weakening. With these, some of the rules and properties of classical logic are dropped or restricted.
The second category is trickier to describe, because we have to be careful with the term 'extension'. In many cases, we cannot extend a logic just by adding axioms or rules. Some logics have the property that they are Post complete meaning that they have no consistent proper extension. This is true of classical propositional logic, for example. But we can create new logics by introducing additional logical operators or connectives and using classical logic as the underlying logic. This is true with the family of modal logics, for example, or temporal logic, or the various conditional logics.
The third category includes connexive logic and abelian logic. These have different rules from classical logic.
So the answer to your question, is: yes we can have non-classical logics that don't break the classical rules, but only by extending classical logic.