The Tate pairing has plenty of attractive applications, e.g., ID-based cryptosystems, short signatures, etc. Recently several fast implementations of the Tate pairing has been reported, which make it appear that the Tate pairing is capable to be used in practical applications. The computation time of the Tate pairing strongly depends on underlying elliptic curves and definition fields. However these fast implementation are restricted to supersingular curves with small MOV degrees. In this paper we propose several improvements of computing the Tate pairing over general elliptic curves over finite fields double-struck Fq (q = pm, p > 3) - some of them can be efficiently applied to supersingular curves. The proposed methods can be combined with previous techniques. The proposed algorithm is specially effective upon the curves that has a large MOV degreek. We develop several formulas that compute the Tate pairing using the small number of multiplications over double-struck Fqk. For k = 6, the proposed algorithm is about 20% faster than previously fastest algorithm.