proof writing - On provability within minimal logic

In its most naive form my question boils down to this:

when is a proposition that is provable "by contradiction" also provable "directly"?

IOW, is it possible to know, a priori, that a proposition $p$, if it's provable at all, it can only be proved by contradiction (and therefore that it would be futile to attempt a direct proof)?

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Let me try to sharpen my question slightly, by using the terminology and definitions given in this MSE answer. Let $C$ be the set¹ of all propositions that are provable using classical logic. Similarly, let $M$ be all the propositions that are provable using minimal logic. Then, according to the cited post, $M \subsetneq C$ ("there are statements that are provable by contradiction that are not provable directly"), or IOW, $C\backslash M \neq \varnothing$.

So my question can be rephrased like this: given some proposition $p \in C$, is it possible to know, a priori (i.e. without necessarily finding a "direct proof" of $p$), whether $p \in M\;$?

In my very limited experience, proofs of results like "$C\backslash M \neq \varnothing$" are very unsatisfying because they rarely illuminate the question of whether a particular $p$ of interest belongs to $M$ or to $C\backslash M$. Is there any theorem from the standard mathematical curriculum but outside of mathematical logic that is known to belong to $C\backslash M$.

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_{¹ or class, or family, or whatever works!}