For all integers $k,d$ such that $k geq 3$ and $k/2leq d leq k-1$, let $n$ be a sufficiently large integer {rm(}which may not be divisible by $k${rm)} and let $sle lfloor n/krfloor-1$. We show that if $H$ is a $k$-uniform hypergraph on $n$ vertices with $delta_{d}(H)>binom{n-d}{k-d}-binom{n-d-s+1}{k-d}$, then $H$ contains a matching of size $s$. This improves a recent result of Lu, Yu, and Yuan and also answers a question of Kuhn, Osthus, and Townsend. In many cases, our result can be strengthened to $sleq lfloor n/krfloor$, which then covers the entire possible range of $s$. On the other hand, there are examples showing that the result does not hold for certain $n, k, d$ and $s= lfloor n/krfloor$.
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