think there is nothing more to do here, closing.

MWE:

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}

\begin{document}
Look at $\hat u$.
Let's see $\widehat{\mathcal F}=\hat F=\bar F=\tilde F=\overline F$.
\end{document}

Compile with tex4ebook with parameter new-accents, accent-, the accents in the second line are rendered well. However, the accent $\hat u$ is rendered incorrectly. I look at the code of html4.4ht, seemingly AEIOUaeiou are treated differently than other alphabets (say F in the seond line). I guess that this is the reason for the incorrect rendering:

\:CheckOption{accent-} \if:Option
\Configure{HAccent}\acute{AEIOUYaeiouy{}}{\Picture+{}}{\EndPicture}
\Configure{HAccent}\bar{}{\Picture+{}}{\EndPicture}
\Configure{HAccent}\breve{}{\Picture+{}}{\EndPicture}
\Configure{HAccent}\check{}{\Picture+{}}{\EndPicture}
\Configure{HAccent}\ddot{AEIOUYaeiouy{}}{\Picture+{}}{\EndPicture}
\Configure{HAccent}\dot{}{\Picture+{}}{\EndPicture}
\Configure{HAccent}\grave{AEIOUaeiou{}}{\Picture+{}}{\EndPicture}
\Configure{HAccent}\hat{AEIOUaeiou{}}{\Picture+{}}{\EndPicture}
\Configure{HAccent}\tilde{AOaoNn{}}{\Picture+{}}{\EndPicture}
\Configure{HAccent}\vec{}{\Picture+{}}{\EndPicture}
\Configure{HAccent}\widehat{}{\Picture+{}}{\EndPicture}
\Configure{HAccent}\widetilde{}{\Picture+{}}{\EndPicture}
\fi