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Spin precession is a key feature in the dynamics of black-hole binaries predicted by General Relativity. Misalignments between the black-hole spins and the binary’s orbital momentum induce characteristic modulations to the emitted gravitational waves which are intrinsically hard to estimate because they provides a highly subdominant contribution. The most commonly used quantity to track the amount of relativistic precession in current LIGO/Virgo observations is the so-called effective precession parameter χ_p. Here we exploit a recently developed re-definition of χ_p that considers all the variations occurring on the precession timescale, allowing to capture two-spin effects in a consistent fashion. In particular, sources with two precessing spins populate a dedicated region of the parameter space where χ_p>1. Using a large number of software injections in synthetic data, we recover the posterior distribution of such augmented χ_p parameter and identify the regions of the parameter space where present and future gravitational-wave interferometers could detect two-spin effects in black-hole binary data.