We present results from the first radio observations of a complete orbit (∼17 d) of the neutron star X-ray binary Circinus X-1 using the Australia Telescope Compact Array Broadband Backend, taken while the system was in an historically faint state. We have captured the rapid rise and decline of a periastron passage flare, with flux densities for 9 d prior to the event stable at ∼1 mJy at 5.5 GHz and ∼0.5 mJy at 9 GHz. The highest flux densities of 43.0 ± 0.5 mJy at 5.5 GHz and 29.9 ± 0.6 mJy at 9 GHz were measured during the flare’s decline (MJD 55206.69) which continues towards pre-flare flux densities over the following 6 d. Imaging of pre-flare data reveals steady structure including two stable components within 15 arcsec of the core which we believe may be persistent emission regions within the system’s outflows, one of which is likely associated with the system’s counter jet. Unlike past observations carried out in the system’s brighter epochs, we observe no significant structural variations within ≈3 arcsec of the core’s position. Model subtraction and difference mapping provide evidence for variations slightly further from the core: up to 5 arcsec away. If related to the observed core flare, then these variations suggest very high outflow velocities with Γ > 35, though this can be reduced significantly if we invoke phase delays of at least one orbital period. Interestingly, the strongest structural variations appear to the north-west of the core, opposite to the strongest arcsec-scale emission historically. We discuss the implications of this behaviour, including the possibility of precession or a kinked approaching jet.