We generalize an efficient hyper-regularity and unimodularity test from differential Ore polynomial matrices to arbitrary Ore polynomial matrices. The core of the contribution consists of algorithms for unimodular row and column completions of arbitrary univariate Ore polynomial matrices, respectively. After a possible degree reduction using noncommutative companion matrices, this is done by a systematic projection of suitable coordinates with a subsequent elimination. Based on previous work for differential Ore polynomial matrices, we remove rather restrictive conditions, which now allows for the application of this algorithm to a larger class of systems that may contain pure algebraic equations.