The irreducible complex representations of the  simply connected simple group $G=Sp_{r,{\mathbb C}}$ of type $C_r$,
for $r>1$, of dimension $n<{\rm dim}\ G$
are listed in the [paper of Andreev, Vinberg, and Elashvili](http://link.springer.com/content/pdf/10.1007%2FBF01076005),
Table 1 (see also the [Russian version](http://www.mathnet.ru/links/419af2a1d9d33839a49ff8898866b056/faa2839.pdf)). 
They are the fundamental irreducible representations $R(\pi_1)$ of dimension $2r$, $R(\pi_2)$ of dimension $2r^2-r-1$,
and, for $r=3$, $R(\pi_3)$ of dimension 14. 
For all $r\ge 2$, $r\neq 3$, we have ${\rm dim}\ R(\pi_1)=2r<2r^2-r-1={\rm dim}\ R(\pi_2)$, 
hence $R(\pi_2)$ is the nontrivial irreducible representation of second smallest dimension.
For $r=3$,  as Jim Humphreys noted, the dimensions are $6,14,14$, so ${\rm dim}\ R(\pi_2)={\rm dim}\ R(\pi_3)>{\rm dim}\ R(\pi_1)$, and 
$R(\pi_2)$ is *a* nontrivial irreducible representation of second smallest dimension.