The rule of addition is correct in classical logic, which is presumably what you are learning. Classical logic is usually understood as the logic that is truth-preserving, i.e. for a valid argument it requires that if the premises of an argument are true then the conclusion follows by necessity, or that it is impossible for the premises to be true and the conclusion false. In practice, this can be quite a weak criterion. It does not require that the premises are relevant to the conclusion at all, and it allows for the principle of explosion, which is that from a contradiction anything follows, and also the principle of implosion, which is that a logical truth follows from any premises whatever.
In simple cases this need not seem odd. If I have a coin in my left pocket, it follows that I have a coin in my left pocket or a coin in my right pocket. But it can definitely seem odd in cases where the introduced disjunction is completely irrelevant. It's just a feature of classical logic. The important thing about it is that it is a truth-preserving move. It will never be the case that A is true and "A or B" is false.
There are other logics that impose different conditions on what qualifies as a valid argument, or which can be understood as preserving something other than just truth. There is a family of logics that go by the name 'relevance logic' that either lack the addition rule, or impose restrictions on it, precisely to block the move from A to "A or B", and thence to "if not A then B". Some of these distinguish two kinds of disjunction, an extensional one that allows the move from A to "A or B", and a separate intensional one that is required to go from "A or B" to "if not A then B".