You might want to use \insertsubsection instead of \insertsubsectionnavigation{⟨width⟩} which according to the manual
Inserts a vertical navigation bar containing all subsections
of the current section, with the current subsection highlighted.
This bar adds additional space to the left and right of the (sub)section. It works the same way with \insertsection versus \insertsectionnavigation{⟨width⟩}.
Please compare with this example:
\documentclass{beamer}
\makeatletter
\setbeamertemplate{headline}{%
\begin{beamercolorbox}[wd=.24\paperwidth,ht=\@tempdimb]{section in head/foot}%
\vbox to\@tempdimb{\vfill\insertsection\vfill\hfill}%
\end{beamercolorbox}%
\begin{beamercolorbox}[wd=.24\paperwidth,ht=\@tempdimb]{section in head/foot}%
\vbox to\@tempdimb{\vfil\insertsubsection\vfil}%
\end{beamercolorbox}%
\begin{beamercolorbox}[wd=.038\paperwidth,ht=\@tempdimb]{section in head/foot}%
\vbox to\@tempdimb{\vfil\hspace*{0.6cm}\rule{2cm}{0.5cm}\vfil}%
\end{beamercolorbox}%
}
\makeatother
\begin{document}
\section{A section}
\subsection{A subsection}
\begin{frame}{A show frame}
just text here
\end{frame}
\end{document}
Comments:
Now that the text directly starts at the frame border, you might want to insert a \hspace*{1em}.
I removed \hskip-17ex in order to render the text within the beamercolorbox and used a rule instead of the UT image.
Update:
Alternatively, you can redefine the templates of \insertsectionnavigationand \insertsubsectionnavigation from beamerbasenavigation.sty for a reduction of the added horizontal space and \RaggedRight text (using the ragged2e package).
\usepackage{ragged2e}
\makeatletter
\renewcommand\insertsectionnavigation[1]{%
\hbox to #1{%
\vbox{{\usebeamerfont{section in head/foot}\usebeamercolor[fg]{section in head/foot}%
\vskip0.5ex%
\def\slideentry##1##2##3##4##5##6{}%
\def\sectionentry##1##2##3##4##5{%
\ifnum##5=\c@part%
\def\insertsectionhead{##2}%
\def\insertsectionheadnumber{##1}%
\def\insertpartheadnumber{##5}%
\setbox\beamer@tempbox=\hbox{\RaggedRight% added for flushed left text
\hyperlink{Navigation##3}{\hbox to #1{%
\hskip0.5ex% adjusted from 0.3 cm
\ifnum\c@section=##1%
{\usebeamertemplate{section in head/foot}}%
\else%
{\usebeamertemplate{section in head/foot shaded}}%
\fi\hskip0.5ex}}}% adjusted from 0.3 cm
\ht\beamer@tempbox=1.6875ex\dp\beamer@tempbox=0.75ex%
\box\beamer@tempbox\fi}%
\dohead\vskip0.5ex}}\hfil}}
\renewcommand\insertsubsectionnavigation[1]{%
\hbox to #1{%
\vbox{{%
\usebeamerfont{subsection in head/foot}\usebeamercolor[fg]{subsection in head/foot}%
\vskip0.5625ex%
\beamer@currentsubsection=0%
\def\sectionentry##1##2##3##4##5{}%
\def\slideentry##1##2##3##4##5##6{\ifnum##6=\c@part\ifnum##1=\c@section%
\ifnum##2>\beamer@currentsubsection%
\beamer@currentsubsection=##2%
\def\insertsubsectionhead{##5}%
\def\insertsectionheadnumber{##1}%
\def\insertsubsectionheadnumber{##2}%
\def\insertpartheadnumber{##6}%
\setbox\beamer@tempbox=\hbox{\RaggedRight\beamer@link(##4){% added for flushed left text
\hbox to #1{\hskip0.5ex% adjusted from 0.3 cm
\ifnum\c@subsection=##2%
{\usebeamertemplate{subsection in head/foot}}%
\else%
{\usebeamertemplate{subsection in head/foot shaded}}%
\fi\hfil\hskip0.5ex}}}% adjusted from 0.3 cm
\ht\beamer@tempbox=1.6875ex\dp\beamer@tempbox=0.75ex%
\box\beamer@tempbox\fi\fi\fi}%
\dohead\vskip0.5625ex}\hfil}}}
% headline template
\setbeamertemplate{headline}{%
\begin{beamercolorbox}[wd=.24\paperwidth]{section in head/foot}%
\insertsectionnavigation{.22\paperwidth}%
\end{beamercolorbox}%
\begin{beamercolorbox}[wd=.24\paperwidth]{section in head/foot}%
\insertsubsectionnavigation{.18\paperwidth}%
\end{beamercolorbox}%
\begin{beamercolorbox}[wd=.038\paperwidth,ht=\@tempdimb]{section in head/foot}%
\vbox to\@tempdimb{\vfil\hspace*{0.6cm}\rule{2cm}{0.5cm}\vfil}%
\end{beamercolorbox}%
}
\makeatother