We focus on the existence of large linear structures within the sets of hypercyclic and frequently hypercyclic vectors. For operators $T$ satisfying Kitai's Criterion or the Frequent Hypercyclicity Criterion, we analyze the fundamental linear space $\{f(T)x | f \in H(\mathbb{C})\}$, studied by Herrero, Bourdon, Bès, Wengenroth, and many others. We show that the set $\{f(T)x | f \in H(\mathbb{C})\}$ can be extended within $HC(T) \cup \{0\}$ or $FHC(T) \cup \{0\}$ if $x \in HC(T)$ or $x \in FHC(T)$, respectively. The extension is such that the quotient of the new space with $\{ f(T)x \mid f \in H(\mathbb{C}) \}$ has dimension $\mathfrak{c}$ (the cardinality of the continuum). Second, we prove that generically a finite-dimensional subspace contained in $HC(T) \cup \{0\}$ can be enlarged to a subspace of dimension $\mathfrak{c}$. Third, we establish sufficient conditions for extending arbitrary linear subspaces both from $HC(T) \cup \{0\}$ and $FHC(T) \cup \{0\}$ to larger subspaces of dimension $\mathfrak{c}$.