Orientable sequences, periodic sequences in which any $n$-tuple appears at most once in either direction, were introduced in the early 1990s for use in certain position location applications; constructions and upper bounds on the period for the binary case were published by Dai et al. More recent work has focussed on $k$-ary sequences for arbitrary $k>2$; one method of construction involves negative orientable sequences, in which an $n$-tuple appears at most once in either the sequence or the negative of its reverse. In this paper we show how additional $n$-tuples can be added to one previously described approach for generating negative orientable sequences, resulting in new sequences with asymptotically optimal period. These sequences can in turn be used to generate orientable sequences, again with asymptotically optimal period.