The majority of existing speech bandwidth extension (BWE) methods operate under the constraint of fixed source and target sampling rates, which limits their flexibility in practical applications. In this paper, we propose a multi-stage speech BWE model named MS-BWE, which can handle a set of source and target sampling rate pairs and achieve flexible extensions of frequency bandwidth. The proposed MS-BWE model comprises a cascade of BWE blocks, with each block featuring a dual-stream architecture to realize amplitude and phase extension, progressively painting the speech frequency bands stage by stage. The teacher-forcing strategy is employed to mitigate the discrepancy between training and inference. Experimental results demonstrate that our proposed MS-BWE is comparable to state-of-the-art speech BWE methods in speech quality. Regarding generation efficiency, the one-stage generation of MS-BWE can achieve over one thousand times real-time on GPU and about sixty times on CPU.