Let (N,e) be a public key of the RSA cryptosystem, and d be the corresponding private key. In practice, we usually choose a small e for quick encryption. In this paper, we improve partial private key exposure attacks against RSA with a small public exponent e. The key idea is that under such a setting we can usually obtain more information about the prime factor of N and then by solving a univariate modular polynomial with Coppersmith's method, N can be factored in polynomial time. Compared to previous results, we reduce the number of d's leaked bits needed to mount the attack by log_2 (e) bits. Furthermore, our experiments show that for 1024-bit N, our attack can achieve the theoretical bound on a personal computer, which verified our attack.