  We calculate a measure of superposition macroscopicity $\mathcal{M}$ for a
superposition of screening current states in a superconducting flux qubit
(SFQ), by relating $\mathcal{M}$ to the action of an instanton trajectory
connecting the potential wells of the flux qubit. When a magnetized
Bose-Einstein condensed (BEC) gas containing $N_{B}\sim \mathcal{O}(10^6)$
atoms is brought into a $\mathcal{O}(1)$ $\mu\text{m}$ proximity of the flux
qubit in an experimentally realistic geometry, we demonstrate the appearance of
a two- to five-fold amplification of $\mathcal{M}$ over the bare value without
the BEC, by calculating the instantion trajectory action from the
microscopically derived effective flux Lagrangian of a hybrid quantum system
composed of the flux qubit and a spin-$F$ atomic Bose gas. Exploiting the
connection between $\mathcal M$ and the maximal metrological usefulness of a
multimode superposition state, we show that amplification of $\mathcal{M}$ in
the ground state of the hybrid system is equivalent to a decrease in the
quantum Cram\'{e}r-Rao bound for estimation of an externally applied flux. Our
result therefore demonstrates the increased usefulness of the BEC--SFQ hybrid
system as a sensor of ultraweak magnetic fields below the standard quantum
limit.
