Variant of Analytic form of Hahn-Banach

Question

The HB theorem states that given a linear functional $f: G \to \mathbb{R}$ where $G$ is a subspace of a normed vector space $E$ and a function $p:E \to \mathbb{R}$ satisfying

1. $p(x+y) \leq p(x)+p(y)$.
2. $p(ax) = ap(x), a > 0$.

Such that $f(x) \leq p(x)$ for all $x \in G$, then $f$ may be extended to a linear functional $\hat{f}$ on $E$ and $\hat{f}(x) \leq p(x)$ for all $x \in E$.

My question is, if $|f(x)| \leq p(x)$, can we find an extension $\hat{f}$ such that $|\hat{f}(x)| \leq p(x)$ for all $x \in E$ without requiring that $p(x) = p(-x)$.