The author introduces here a new definition of strong almost summability. In contrast to an earlier alternative definition given by the author in J. Austral. Math. Soc., 22 A, 446-455 (1976; Zbl 0346.46006) the present definition is capable of covering the definition of strong almost convergence introduced by \textit{I. J. Maddox} [Math. Proc. Camb. Philos. Soc. 83, 61-64 (1978; Zbl 0392.40001)]. Certain properties of the spaces of strongly almost summable sequences have been studied. In J. Indian Math. Soc., 40, 173-184 (1976; Zbl 0445.40003) the author has extended the spaces \(\hat c\) of all almost convergent sequences and \(\hat c_ 0\) of all sequences which are almost-convergent to zero to \(\hat c(p)\) and \(\hat c_ 0(p)\). The author generalises these spaces by using infinite matrices.