You can clip the symbol:
\documentclass{article}
\usepackage{amssymb}
\usepackage{trimclip}
\makeatletter
\DeclareRobustCommand{\leftloop}{%
\mathrel{\mathpalette\left@loop\relax}%
}
\newcommand{\left@loop}[2]{%
\vphantom{\looparrowright}
\smash{\clipbox{0 {-.1\height} {.35\width} {-.1\height}}{$\m@th#1{\looparrowright}$}}%
}
\makeatother
\begin{document}
$a\leftloop b_{c\leftloop d}$
\end{document}

Code borrowed from https://tex.stackexchange.com/a/395049/4427
This means “cut nothing on the left, add a negative clip on the bottom, clip 35% of the width from the right, add a negative clip on the top”. Why negative clipping? Because the glyph slightly overshoots its bounding box. So I also smash the clipped symbol and set the height with the help of \vphantom.
Can we keep the round cap? Yes, by adding a clipped minus sign.
\documentclass{article}
\usepackage{amssymb}
\usepackage{trimclip}
\makeatletter
\DeclareRobustCommand{\leftloop}{%
\mathrel{\mathpalette\left@loop\relax}%
}
\newcommand{\left@loop}[2]{%
\smash{\clipbox{0 {-.1\height} {.4\width} {-.1\height}}{$\m@th#1{\looparrowright}$}}%
\clipbox{{0.8\width} 0 0 0}{$\m@th#1-$}%
\mkern-1mu
}
\makeatother
\begin{document}
\fboxrule=0.1pt\fboxsep=0pt\fbox{$\leftloop$}% just to see the bounding box
$a\leftloop b_{c\leftloop d}$
\end{document}

No need to add a phantom, because the minus sign has the right height and depth.
\hookrightarrowis composed with two gliphs, but\looprightarrowisn't. You can clip the symbol, though. – egreg Jul 08 '20 at 15:00\DeclareMathSymbol{\looparrowright} {\mathrel}{AMSa}{"23}found inamssymb.dtxat https://ctan.org/tex-archive/fonts/amsfonts. No idea what that means, though. – Dr. Manuel Kuehner Jul 08 '20 at 15:03