Standard tableaux of skew shape are fundamental objects in enumerative and algebraic combinatorics and no product formula for the number is known. In 2014, Naruse gave a formula (NHLF) as a positive sum over excited diagrams of products of hook-lengths. Subsequently, Morales, Pak, and Panova gave a $q$-analogue of this formula in terms of skew semistandard tableaux (SSYT). They also showed, partly algebraically, that the Hillman--Grassl map, restricted to skew semistandard tableaux, is behind their $q$-analogue. We study the problem of circumventing the algebraic part and proving the bijection completely combinatorially, which we do for border strips. For a skew shape, we define minimal semistandard Young tableaux, that are in correspondence with excited diagrams via a new description of the Hillman--Grassl bijection and have an analogue of excited moves. Lastly, we relate the minimal skew SSYT with the terms of the Okounkov-Olshanski formula (OOF) for counting standard tableaux of skew shape. Our construction immediately implies that the summands in the NHLF are less than the summands in the OOF and we characterize the shapes where both formulas have the same number of summands.