This paper proposes a method for improved CTC inference with searched intermediates and multi-pass conditioning. The paper first formulates self-conditioned CTC as a probabilistic model with an intermediate prediction as a latent representation and provides a tractable conditioning framework. We then propose two new conditioning methods based on the new formulation: (1) Searched intermediate conditioning that refines intermediate predictions with beam-search, (2) Multi-pass conditioning that uses predictions of previous inference for conditioning the next inference. These new approaches enable better conditioning than the original self-conditioned CTC during inference and improve the final performance. Experiments with the LibriSpeech dataset show relative 3%/12% performance improvement at the maximum in test clean/other sets compared to the original self-conditioned CTC.