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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 Belleexperiment 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 greaterthan $5.3\sigma$ and $4.0\sigma$, respectively. We also investigate the energydependence of the $e^+e^-\to\pi^+\pi^-\pi^0\chi_{bJ}$ cross section, but wecannot determine whether the contributions are from the $\Upsilon(10860)$ and$\Upsilon(11020)$ resonances or non-resonant continuum processes. The signalsfor $e^+e^-\to \phi\chi_{bJ}$ are not significant, and the upper limits of theBorn 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-massenergies from 10.96 to 11.05~GeV.

Observation of $e^+e^-\to\pi^+\pi^-\pi^0\chi_{b1,2}(1P)$ and search for $e^+e^-\to\phi\chi_{b1,2}(1P)$ at $\sqrt{s}=$10.96---11.05 GeV