An alternative to the traditional Büchi Automata (BA), called Testing Automata (TA) was proposed by Hansen et al. [8,6] to improve the automata-theoretic approach to LTL model checking. In previous work [2], we proposed an improvement of this alternative approach called TGTA (Generalized Testing Automata). TGTA mixes features from both TA and TGBA (Generalized Büchi Automata), without the disadvantage of TA, which is the second pass of the emptiness check algorithm. We have shown that TGTA outperform TA, BA and TGBA for explicit and symbolic LTL model checking. However, TA and TGTA are less expressive than Büchi Automata since they are able to represent only stutter-invariant LTL properties (LTL ∖X) [13]. In this paper, we show how to extend Generalized Testing Automata (TGTA) to represent any LTL property. This allows to extend the model checking approach based on this new form of testing automata to check other kinds of properties and also other kinds of models (such as Timed models). Implementation and experimentation of this extended TGTA approach show that it is statistically more efficient than the Büchi Automata approaches (BA and TGBA), for the explicit model checking of LTL properties.