We study positive partially localized solutions of the elliptic equation \begin{equation}\tag{1} \Delta_x u+u_{yy}+f(u)=0,\quad (x,y)\in\mathbb{R}^N\times\mathbb{R}, \end{equation} where $N\geq 2$ and $f$ is a $C^1$ function satisfying $f(0)=0$ and $f'(0)<0$. By partially localized solutions we mean solutions $u(x,y)$ which decay to zero as $|x|\to\infty$ uniformly in $y$.  Our main concern is the existence of positive partially localized solutions which are quasiperiodic in $y$. The fact that such  solutions can exist in equations of the above form was demonstrated in our earlier work: we proved that the nonlinearity $f$ can be designed in such a way that equation (1) possesses positive partially localized quasiperiodic solutions with 2  frequencies. Our main contributions in the present paper are twofold. First, we improve the previous result by showing that positive partially localized quasiperiodic solutions with any prescribed number $n\ge 2$ of frequencies exist for some nonlinearities $f$. Second, we give a tangible sufficient condition on  $f$ which guarantees that equation (1) has such quasiperiodic solutions, possibly after $f$ is perturbed slightly.  The condition, with $n=2$, applies, for example, to some combined-powers nonlinearities  $f(u)=u^p+\lambda u^q-u$ with suitable exponents $p>q>1$ and coefficient $\lambda>0$.