Private comparison protocols are fundamental to the field of secure computation. Recently, Lu et al. (ASIACCS 2018) proposed a new protocol, XCMP,, which is based on a ring-based fully homomorphic encryption (FHE) scheme. In that scheme, two μ-bit integers a and b are compared in encrypted form without revealing the plaintext to an evaluator. The protocol outputs a bit in encrypted form, which indicates whether a > b. XCMP has the following three advantages: the output can be reused for further processing, the evaluation is performed without any interactions with a decryptor having a secret key, and the required multiplicative depth is only 1. However, XCMP has two potential disadvantages. First, the protocol result preserves both additive and multiplicative homomorphisms over ℤ t only, whereas the underlying FHE scheme can support a much larger plaintext space of (Formula Presented) for a prime t and a power-of-two N; this restricts the functionality of applications using the comparison result. Second, the bit length μ of the integers to be compared is no more than log N (typically 16 bits, at most). Thus, it is difficult for XCMP to handle larger integers. In this paper, we propose a non-interactive private comparison protocol that solves the aforementioned problems and outputs an additively and multiplicatively reusable comparison result over the ring without adding an extremely large computational overhead over XCMP. Moreover, by regarding a μ (>16 -bit integer as a sequence of chunks, we show that the multiplicative depth required for our comparison protocol is logarithmic in the number of chunks. This value is much smaller than the naïve solution with a multiplicative depth of log μ. Experiment results demonstrate that our protocol introduces a subtle overhead over XCMP. Remarkably, we experimentally demonstrate that our protocol for a larger domain is comparable to the construction given by one of the state-of-the-art bitwise FHE schemes.