  The microscopic origin of $1/f$ magnetic flux noise in superconducting
circuits has remained an open question for several decades despite extensive
experimental and theoretical investigation. Recent progress in superconducting
devices for quantum information has highlighted the need to mitigate sources of
qubit decoherence, driving a renewed interest in understanding the underlying
noise mechanism(s). Though a consensus has emerged attributing flux noise to
surface spins, their identity and interaction mechanisms remain unclear,
prompting further study. Here we apply weak in-plane magnetic fields to a
capacitively-shunted flux qubit (where the Zeeman splitting of surface spins
lies below the device temperature) and study the flux-noise-limited qubit
dephasing, revealing previously unexplored trends that may shed light on the
dynamics behind the emergent $1/f$ noise. Notably, we observe an enhancement
(suppression) of the spin-echo (Ramsey) pure dephasing time in fields up to
$B=100~\text{G}$. With direct noise spectroscopy, we further observe a
transition from a $1/f$ to approximately Lorentzian frequency dependence below
10 Hz and a reduction of the noise above 1 MHz with increasing magnetic field.
We suggest that these trends are qualitatively consistent with an increase of
spin cluster sizes with magnetic field. These results should help to inform a
complete microscopic theory of $1/f$ flux noise in superconducting circuits.
