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In an entanglement swapping scenario, if two sources sharing entangled states between three parties are independent, local correlations lead to a different kind of inequalities than the standard Bell inequalities, known as network local models. A highly demanding task is to find out a way to involve many players nontrivially in a quantum network since measurements, in general, disturb the system. To this end, we consider here a novel way of sharing network nonlocality when two observers initially share close to a maximally entangled states. We report that by employing unsharp measurements performed by one of the observers, six pairs can sequentially demonstrate the violation of bilocal correlations while a maximum of two pairs of observers can exhibit bi-nonlocality when both the observers perform unsharp measurements. We also find the critical noise involved in unsharp measurements in each round to illustrate the bi-nonlocality for a fixed shared entangled state as a resource. We also establish a connection between entanglement content of the shared state, quantified via von-Neumann entropy of the local density matrix for pure states and entanglement of formation for Werner states, and the maximum number of rounds showing violation of bilocal correlations. By reducing entanglement content in the elements of the joint measurement by the third party, we observe that the maximum number reduces to two sequential sharing of bi-nonlocality even for the maximally entangled state when the settings at each side are taken to be three and fixed.