<p>A (1,&#xa0;<i>k</i>)-overlap-free code of length <i>n</i> over <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {Z}_q\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> is a non-empty subset of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {Z}_q^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">Z</mi> <mi>q</mi> <mi>n</mi> </msubsup> </math></EquationSource> </InlineEquation> in which the prefix set with length at most <i>k</i> of each codeword does not coincide with the suffix of the same codeword or any other codeword. Such family of codes has applications in DNA-based storage systems. In this paper, extending the Zero Block Construction proposed by Blackburn et al., we exhibit a new family of <i>q</i>-ary (1,&#xa0;<i>k</i>)-overlap-free codes. We then establish a necessary and sufficient condition for the codes to be non-expandable. For the expandable codes, we construct a new family of <i>q</i>-ary (1,&#xa0;<i>k</i>)-overlap-free codes to expand the codes to be non-expandable. Furthermore, enumeration formulas and lower bounds for the size of the resulting codes are provided, and comparisons with the related works are also given.</p>