The concept of threshold public key encryption (TPKE) with the special property called key re-splittability (re-splittable TPKE, for short) was introduced by Hanaoka et al. (CT-RSA 2012), and used as one of the building blocks for constructing their proxy re-encryption scheme. In a re-splittable TPKE scheme, a secret key can be split into a set of secret key shares not only once, but also multiple times, and the security of the TPKE scheme is guaranteed as long as the number of corrupted secret key shares under the same splitting is smaller than the threshold. In this paper, we show several new constructions of a re-splittable TPKE scheme by extending the previous (ordinary) TPKE schemes. All of our proposed schemes are based on discrete logarithm (DL)-type assumptions. Therefore, our results suggest that key re-splittability is a very natural property for DL-type TPKE schemes.