Disclaimer: I am not an expert on this topic and only a fellow student of ZK protocols. Most of this answer is based on the article "Sorting out zero-knowledge", mentioned below, and corrections to this answer is very welcome.
I think what are you seeking is systems with the stronger notion of zero-knowledge with auxiliary input¹, which states that
"no matter which additional input is given to the verifier before the interaction with the prover, the interaction does not help the verifier learn anything at all that he could not have learnt by himself given the same additional input".²
(The additional input considering your scenario would be the information in you got in previous proofs)
1: Y. Oren, "On the cunning power of cheating verifiers: Some observations about zero knowledge proofs," 28th Annual Symposium on Foundations of Computer Science (sfcs 1987), Los Angeles, CA, USA, 1987, pp. 462-471.
2: Brassard, Gilles, and Claude Crepeau. "Sorting out zero-knowledge." Advances in Cryptology—EUROCRYPT’89: Workshop on the Theory and Application of Cryptographic Techniques Houthalen, Belgium, April 10–13, 1989 Proceedings 8. Springer Berlin Heidelberg, 1990.
Regarding your comment "But does it make sense to keep calling them "zero knowledge proofs" then? It feels misleading."
This question was also raised by Feige, Fiat and Shamir in their paper "Zero-knowledge proofs of identity". There, they call what you refer as ZK proof as a ZK proof of assertion, and they propose a definition of ZK where the prover prove that he knows the status of an information I with respect to a language L instead of proving that I belongs to L. In their own words:
As a motivating (but technically inaccurate) example, consider a prover A who
wants to prove to a skeptical B that he has settled Fermat's last theorem. With the type of proof introduced in this paper, A can convince B that he is a mathematical superstar without telling B anything new about the problem--not even whether he has found a proof or a counterexampte!
On the same vein, Galil, Haber and Yung also suggested that ZK proofs should be termed instead as proofs of minimum knowledge, meaning that they only reveal what is necessary for them to fulfill their intended purpose.