  Association schemes play an important role in algebraic combinatorics and
have important applications in coding theory, graph theory and design theory.
The methods to construct association schemes by using bent functions have been
extensively studied. Recently, in [13], {\"O}zbudak and Pelen constructed
infinite families of symmetric association schemes of classes $5$ and $6$ by
using ternary non-weakly regular bent functions.They also stated that
constructing $2p$-class association schemes from $p$-ary non-weakly regular
bent functions is an interesting problem, where $p>3$ is an odd prime. In this
paper, using non-weakly regular bent functions, we construct infinite families
of symmetric association schemes of classes $2p$, $(2p+1)$ and $\frac{3p+1}{2}$
for any odd prime $p$. Fusing those association schemes, we also obtain
$t$-class symmetric association schemes, where $t=4,5,6,7$. In addition, we
give the sufficient and necessary conditions for the partitions $P$, $D$, $T$,
$U$ and $V$ (defined in this paper) to induce symmetric association schemes.
