You do not need Hahn-Banach to extend a continuous linear map defined in a *dense* subspace of a Hausdorff topological vector space into another, *complete* Hausdorff topological vector space. The extension is *uniquely* defined by (uniform) continuity and even satisfies the same (semi)norm bounds.

In raw detail, one simply defines the image of the limit of a Cauchy net in the dense subspace as the limit of the image of that net. Uniform continuity ensures that this new net is also Cauchy, and continuous linear maps are always uniformly continuous (with respect to the uniform structures induced by the vector topologies in the domain and codomain). Completeness and the Hausdorff property of the codomain, on their turn, guarantee that the latter limit exists and is unique. It is straightforward to verify that linearity and preservation of (semi)norm bounds by the extension hold.

Because of that, it does not matter if the multiplier representation of $T_m$ only makes sense in the dense subspace $\mathscr{S}$ (endowed with the $L^p$ norm). In fact, even the Fourier transform itself is defined in (say) $L^2$ like this - strictly speaking, the classical Fourier integral formula for $\hat{f}$ does not make sense for a general $f\in L^2$ but only in the dense subspaces $\mathscr{S}$ or $L^2\cap L^1$. Nevertheless, thanks to the Plancherel formula, the *linear map* defined by this formula extends uniquely and continuously to $L^2$ (but *not* the formula itself). The same is true of more general Fourier integral operators.

Since the Fourier transform is a topological isomorphism of $\mathscr{S}'$ onto itself and all $L^p$ spaces embed into $\mathscr{S}'$, one can even turn this reasoning around and *define* $m\hat{f}$ as the Fourier transform of $T_m f$ for $f\in L^p$, $1\leq p\leq\infty$, with $T_m$ uniquely defined by continuous extension from the dense subspace $\mathscr{S}$ through the $L^p$ bounds you wrote. In other words, if $T_m$ is bounded in the $L^p$ norm, it makes sense to define $m\hat{f}$ like this for any $f\in L^p$. The whole point of the multiplier theorems is to single out classes of $m$'s for which this operation makes sense.

**Edit:** the above argument addresses the OP's concern about whether it is needed to make actual sense of $m\hat{f}$ for $f\in L^p$ in order to define $T_m$. That being said, let us now assess the question in the title, which was not done before. Theorem 7.9.3, pp. 242 of L. Hörmander's book *The Analysis of Linear Partial Differential Operators I - Distribution Theory and Fourier Analysis* (Springer-Verlag, 1983) shows that if $f\in L^p(\mathbb{R}^n)$, then $\hat{f}$ belongs to the (negative-order) $L^2$ Sobolev space $H_{-s}(\mathbb{R}^n)$ for $2<p\leq\infty$ and $s>n(\frac{1}{2}-\frac{1}{p})$, and $H_{-s}$ consists of distributions of order $k$ for $s\leq k\in\mathbb{N}$. Conversely, Theorem 7.6.6, pp 209-210 of the same book shows that if the Fourier transform of $L^p(\mathbb{R}^n)$ consists of distributions of order $k$, then $k\geq n(\frac{1}{2}-\frac{1}{p})$. In other words, there certainly are distributions of order *greater but not smaller* (ahem) *than* $n(\frac{1}{2}-\frac{1}{p})$ which are Fourier transforms of elements of $L^p(\mathbb{R}^n)$ if $p>2$.