  Micron-sized superconducting interference devices ($\mu$-SQUIDs) based on
constrictions optimized for minimizing thermal runaway are shown to exhibit
voltage oscillations with applied magnetic flux despite their hysteretic
behavior. We explain this remarkable feature by a significant supercurrent
contribution surviving deep into the resistive state, due to efficient heat
evacuation. A resistively shunted junction model, complemented by a thermal
balance determining the amplitude of the critical current, describes well all
experimental observations, including the flux modulation of the (dynamic)
retrapping current and voltage by introducing a single dimensionless parameter.
Thus hysteretic $\mu$-SQUIDs can be operated in the voltage read-out mode with
a faster response. The quantitative modeling of this regime incorporating both
heating and phase dynamics paves the way for further optimization of
$\mu$-SQUIDs for nano-magnetism.
