Let \(M\) be a \((2m+r)\)-dimensional Riemannian manifold endowed with an almost \(r\)-contact structure defined by the Reeb vector fields \(\xi_ s\) \((s,t\in \{2m+1,\dots,2m+r\})\) and let \(\eta^ s\) be the associated Reeb covectors, i.e. \(\eta^ t(\xi_ s)= \delta_{t^ s}\). In addition it is assumed that \(M\) is endowed with a (1.1)-structure tensor field \(\Phi\), such that \(\Phi^ 2 X= -X+ \Sigma \eta^ s(X)\xi_ s\), \(\Phi(\xi_ s)= 0\), \(X\in \chi M\). The author defines the almost \(r\)-contact structure \((\Phi,\eta^ s,\xi_ s,g)\) as invariant strictly regular if the 1-forms \(\eta^ s\) are invariant under the action of the Lie group \(G\) generated by \(\xi_ s\). He proves that such a structure induces a principal bundle whose base manifold bears an almost complex structure. More generally the motivation of this paper is to extend some results obtained by \textit{K. Oguie} [Kōdai Math. Sem. Reports 17, 53-62 (1965; Zbl 0136.181)].