  We report searches for the processes $e^+e^-\to\pi^+\pi^-\pi^0\chi_{bJ}$ and
$e^+e^-\to\phi\chi_{bJ}$ (J=1,2) based on data samples collected by the Belle
experiment at the KEKB collider. We report the first observation of the process
$e^+e^-\to (\pi^+\pi^-\pi^0)_{\rm non-\omega}\chi_{b1}$ and first evidence for
$e^+e^-\to\omega\chi_{bJ}$ in the vicinity of the $\Upsilon(11020)$ resonance,
with center-of-mass energies from 10.96 to 11.05~GeV. The significances for
$(\pi^+\pi^-\pi^0)_{\rm non-\omega}\chi_{b1}$ and $\omega\chi_{bJ}$ are greater
than $5.3\sigma$ and $4.0\sigma$, respectively. We also investigate the energy
dependence of the $e^+e^-\to\pi^+\pi^-\pi^0\chi_{bJ}$ cross section, but we
cannot determine whether the contributions are from the $\Upsilon(10860)$ and
$\Upsilon(11020)$ resonances or non-resonant continuum processes. The signals
for $e^+e^-\to \phi\chi_{bJ}$ are not significant, and the upper limits of the
Born cross sections at the 90\% confidence level are $0.7$ and $1.0$~pb for
$e^+e^-\to \phi\chi_{b1}$ and $\phi\chi_{b2}$, respectively, for center-of-mass
energies from 10.96 to 11.05~GeV.
