Air Taylor bubbles in a millichannel filled with water are characterized by an elongated shape, a bullet-shaped nose (head), and a comparatively flat tail. Many experimental and numerical investigations have been performed in the past. Yet, most of them consider Taylor bubbles in a straight channel with constant cross section. The effect of a local change in the channel geometry on both the bubble shape and the flow fields on each side of the gas-liquid interface is, however, difficult to predict. In this work, we present experimental data obtained in a vertical millichannel, where the flow is moderately obstructed by a constriction, whose ratio ranges from 10% to 36%. We find that the Taylor bubble takes an equilibrium position for downward liquid flow with 264.36 < Re < 529.67 and 264.36 < Re < 728.29 for 10.17% and 18.06% constriction ratios, respectively. In this area, an empirical correlation characterizing the bubble head is provided. Other flow regimes, such as bubble breakup, co- and countercurrent configurations, are identified and shown in the form of a regime map. The results, besides their relevance in process engineering, exhibit high reproducibility and will serve as a reference for future interface-resolving two-phase flow simulations.