Complexity of ODD-SMALLER-SAT
Given $\phi$ and an assignment $\overline{y}$, the problem of deciding if there exists a satisfying assignment $\overline{x} \le \overline{y}$ (w.r.t. the lexicographical order) is in NP.
This can be seen by either designing a nondeterministic Turing machine that solves the problem in polynomial time (nondeterministically guess $\overline{x}$, then deterministically check if $\overline{x} \le \overline{y}$ and $\phi(\overline{x}) = \top$, if so accept, otherwise reject), or by noticing that $\overline{x}$ itself is a yes-certificate for the instance (that can be deterministically checked in polynomial-time).

Then your original problem is in $\Delta_2 = P^{NP}$: perform a binary search on the space of the $2^n$ possible assignments to find the smallest satisfying assignment $\overline{x}^*$ for $\phi$. This can be done by repeatedly solving the above problem, each time halving the search space. After $O(\log 2^n) = O(n)$ iterations you are left with $\overline{x}^*$. Check whether $\overline{x}^*_n = 1$ and answer accordingly.