  Motivated by the problem of classifying toric $2$-Fano manifolds, we
introduce a new invariant for smooth projective toric varieties, the minimal
projective bundle dimension. This invariant $m(X)\in\{1, \dots,\dim(X)\}$
captures the minimal degree of a dominating family of rational curves on $X$
or, equivalently, the minimal length of a centrally symmetric primitive
relation for the fan of $X$. We classify smooth projective toric varieties with
$m(X)\geq \dim(X)-2$, and show that projective spaces are the only $2$-Fano
manifolds among smooth projective toric varieties with $m(X)\in\{1,
\dim(X)-2,\dim(X)-1,\dim(X)\}$.
