Analog-to-digital converters (ADCs) are known to suffer from saturation and clipping for inputs exceeding their dynamic range (DR). Recently, the Unlimited Sampling approach addressed this problem by inserting a modulo non-linearity between the input and the ADC. Moreover, a new model called modulo-hysteresis was introduced to enable the recovery of different classes of inputs from noisy observations. ADCs are typically assumed to acquire the instantaneous input amplitude via an inner product with a Dirac delta function. However, a more exact model of the ADC does not satisfy this assumption, and involves computing a local average of the input around the sampling time. Here we address the problem of reconstructing inputs that exceed the ADC DR from local averages. To avoid saturation, we propose a an architecture consisting of a modulo-hysteresis in series with an ADC performing average sampling. We introduce a recovery algorithm and theoretical guarantees for input reconstruction. Moreover, we introduce an additional practical algorithm that works with low sampling rates. Our results are validated by numerical studies using synthetic and hardware data.