  In this paper we present several values for the next-to-minimal weights of
projective Reed-Muller codes. We work over $\mathbb{F}_q$ with $q \geq 3$ since
in IEEE-IT 62(11) p. 6300-6303 (2016) we have determined the complete values
for the next-to-minimal weights of binary projective Reed-Muller codes. As in
loc. cit. here we also find examples of codewords with next-to-minimal weight
whose set of zeros is not in a hyperplane arrangement.
