Grammatical inference involves learning a formal grammar as a finite state machine or a set of rewrite rules. This paper focuses on inferring nondeterministic finite automata (NFA) from a given sample of words: the NFA must accept some words, and reject the others. To address this task we construct several over-constrained inference models capable of finding NFA of size , which are directly convertible to NFA of size . Additionally, we propose an NFA evaluation method based on random walks along with two methods used to enhance the acceptance rates of the inferred NFA. The effectiveness of our approach is demonstrated through the results of comprehensive experiments conducted on several benchmark sets. This paper is an extended version of the ICCS 2023 paper entitled “Inference of over-constrained NFA of size to efficiently and systematically derive NFA of size for grammar learning” (Jastrzab et al., 2023 [1]).