Let $\mathcal{R}$ be an o-minimal expansion of the real field and let $T$ denote the elementary theory of $\mathcal{R}$. Let $\mathcal{H}$ be a Hardy field, that is, an ordered differential field of germs of real-valued unary functions at $+\infty$. Following van den Dries, Macintyre, and Marker, we say that $\mathcal{H}$ is an $\mathcal{R}$-Hardy field if $\mathcal{H}$ is closed under all $\mathcal{R}$-definable functions. In this talk, I will introduce the class of $H_T$-fields as a framework for studying the model theory of $\mathcal{R}$-Hardy fields. The relationship between $\mathcal{R}$-Hardy fields and $H_T$-fields parallels the relationship between Hardy fields and $H$-fields (as studied by Aschenbrenner, van den Dries, and van der Hoeven). I will discuss some partial progress toward finding a model companion for the theory of $H_T$-fields when $T$ is polynomially bounded. I'll also mention an embedding result relating $\mathbb{R}_{\operatorname{an},\exp}$-Hardy fields and the field of logarithmic-exponential transseries.