Quantum magnetometry based on adaptive phase estimation allows for Heisenberg precision while avoiding creation and maintenance of complex entangled states. However, the absolute sensitivity is limited by the nonoptimal use of quantum resources provided by multiple-qubit devices and algorithmic realizations of the protocol. Here, addressing both issues, we advance the time-ascending phase estimation protocol by numerical improvements of Bayesian learning, i.e., sequential updating of the field distribution, and optimal exploitation of resources provided by unentangled qubits with limited coherence. Such algorithmic improvements are used to evaluate the absolute sensitivity both on a simulator and by pulsed-transmon experiments conducted on the IBMQ platform, where we take advantage of high coherence time. In addition, we compare the proficiency of separable and entangled states for magnetometry and show that, in practice, separable states provide superior performance. Flux-sensing emulation experiments demonstrate that a sensitivity of (0.17-1.74)μφ0(Hz)-1 (where φ0 is the flux quantum) for a single-qubit magnetometer and (0.06-0.65)μφ0(Hz)-1 for a five-qubit magnetometer can be achieved for slowly oscillating 1-10kHz magnetic fields, which is comparable to more established experimental platforms for magnetometry.