Fair exchange protocols let two mutually distrustful parties exchange digital data in a way that neither party can cheat. They have various applications such as the exchange of digital items, or the exchange of digital coins and digital services between a buyer and seller. At CCS 2017, two blockchain-based protocols were proposed to support the fair exchange of digital coins and a certain service; namely, "proofs of retrievability" (PoR). In this work, we identify two notable issues of these protocols, (1) waste of the seller's resources, and (2) real-time information leakage. To rectify these issues, we formally define and propose a blockchain-based generic construction called "recurring contingent service payment" (RC-S-P). RC-S-P lets a fair exchange of digital coins and verifiable service occur periodically while ensuring that the buyer cannot waste the seller's resources, and the parties' privacy is preserved. It supports arbitrary verifiable services, such as PoR, or verifiable computation and imposes low on-chain overheads. Also, we present a concrete efficient instantiation of RC-S-P when the verifiable service is PoR. The instantiation is called "recurring contingent PoR payment" (RC-PoR-P). We have implemented RC-PoR-P and analysed its cost. When it deals with a 4-GB outsourced file, a verifier can check a proof in 90 milliseconds, and a dispute between prover and verifier is resolved in 0.1 milliseconds.