Within the framework of the conventional QCD sum rules, we study the pion two-point correlation function, $i\int d^4x e^{iq\cdot x} \langle 0| T J_N(x) {\bar J}_N(0)|\pi(p)\rangle$, beyond the soft-pion limit. We construct sum rules from the three distinct Dirac structures, $i \gamma_5\not\!p$, $i \gamma_5$, $\gamma_5 \sigma_{\mu \nu} {q^\mu p^\nu}$ and study the reliability of each sum rule. The sum rule from the third structure is found to be insensitive to the continuum threshold, $S_\pi$, and contains very small contribution from the undetermined single pole which we denote as $b$. The sum rule from the $i \gamma_5$ structure is also insensitive to $S_\pi$ and $b$ but its result is very different from the one coming from the $\gamma_5 \sigma_{\mu \nu} {q^\mu p^\nu}$ structure. On the other hand, the sum rule from the $i \gamma_5$$\not\!p$ structure has strong dependence on both inputs, which is clearly in constrast with the sum rule for $\gamma_5 \sigma_{\mu \nu} {q^\mu p^\nu}$. We identify the source of the sensitivity for each of the sum rules by making specific models for higher resonance contributions and discuss the implication.