The suspended microchannel resonator is one of the most reliable devices for the mass measurement of biological and chemical specimens. Since it works in vacuum, the energy dissipation is decreased to the minimum. In this case, the instability related to the thermoelastic coupling effect has become a major source of uncertainty and a main hindrance for higher mass precision. In order to explore methods to control the thermoelastic coupling effect, we investigated factors that are relevant to the instability in eigenfrequency measurements, including the thickness of metal coating on the top of the resonator, and the position of the detecting laser spot on the resonator. We found that this instability-induced error is 1 to 2 orders of magnitude lower on resonators without coating than on those with coating, and 3 to 4 orders of magnitude lower if the measurement is done with the first flexural eigenmode than with the first torsional mode. The minimum mass error is achieved when measuring the first flexural eigenmode on an uncoated suspended microchannel resonator. It is in the magnitude of zeptogram (10e-21g), and can be smaller than the minimum detectable mass change which is limited by the minimum detectable frequency change in the measurement.