Airborne acoustic trapping by ultrasonic phased arrays has seen great advances in recent years, and yet the manipulation of objects with different shapes and sizes or heavy particles remains challenging. Here, we demonstrate that the manipulation capabilities of a standing-wave acoustic levitator can be extended by introducing intracavity high-order transverse (HOT) modes in the azimuthal direction, enabling the simultaneous trapping of several objects within a wide range of shapes and sizes with positional and rotational stability, including objects with sizes larger than one wavelength and weights in the scale of millinewtons. The conditions to generate different HOT modes are theoretically analyzed and experimentally implemented. We numerically calculate the pressure distributions, exhibiting good qualitative agreement with the experimental pressure distributions obtained with schlieren images. In addition, we calculate the acoustic force field for several examples of HOT modes and different particle sizes, which leads to a qualitative understanding of the experimental observations.