Quantum metrology enhances measurement precision by utilising the properties of quantum physics. In interferometry, this is typically achieved by evolving highly-entangled quantum states before performing single-shot measurements to reveal information about an unknown phase. While this is the theoretical optimum approach, implementation with all but the smallest states is still extremely challenging. An alternative approach is quantum jump metrology, which deduces information by continuously monitoring an open quantum system while inducing phase-dependent temporal correlations with the help of quantum feedback. Taking this approach here, we analyse measurements of a relative phase in an optical network of two cavities with quantum feedback in the form of laser pulses. It is shown that the proposed scheme can exceed the standard quantum limit without the need for complex quantum states.