This paper proposes an approach for automatically measuring the linewidth-enhancement factor (LEF) of semiconductor lasers using optical feedback self-mixing interferometry (OFSMI), which works in weak optical feedback regime and where the external target is subject to simple harmonic vibration with unknown vibration frequency and magnitude. According to well-known Lang-Kobayashi theory the waveform of the modulated optical output power from the OFSMI system is influenced by multiple parameters, including the LEF, the optical feedback level factor, and the parameters related to the movement of external target. In order to estimate LEF, other parameters must also be considered and, hence, a multiple parameter estimation strategy is required. We propose a solution for this multiple parameter estimation problem based on the principle of data-to-theoretical model match. In particular, a strategy for minimizing a cost function in order to achieve the best fitting is proposed with which all the unknown parameters can be estimated. The performance of the proposed approach is tested using experimental data in comparison with other two approaches. It is seen that, over different experimental signals, the standard deviation for estimated LEF is less than 4.58% on average, which shows that results have excellent consistency. Moreover, the proposed approach also provides a solution for vibration measurement (that is, vibration frequency and magnitude).