Alignment problem in math mode

Your alignment is out as on the even numbered rows you are missing a & so the fragments are right aligned not left, this is shown in the first block, but it appears you want the alignment in the second block (which seems a bit confusing but...)

The negative spacing in the requested form completely obscures the mathematics: the = that are referring to the same value are not aligned, but = referring to different values are aligned.

If the values are too wide to align all the = I would align = that refer to the same value, but allow the individual equations to align separately as in ccc

\documentclass[11pt,letterpaper,twoside]{book}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[total={6in,10in},left=1.5in,top=0.5in,includehead,includefoot]{geometry}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{mathtools}

\begin{document}

aaa
\begin{alignat}{2}
    AAAAA &= BBBBBB \\
    &&&= CCCCCC + DDDDDDD,
\\[1ex]
    EEE &= FFFFFFF \\
    &&&= 38 \, GGGGGGGG + 45 \, HHHHHHH,
\\[1ex]
    KKKKKKKKK &= LL \\
    &&&= MM + N.
\end{alignat}


bbb
\begin{alignat}{1}
    AAAAA &= BBBBBB \\
    &\hspace{-1cm}= CCCCCC + DDDDDDD,
\\[1ex]
    EEE &= FFFFFFF \\
    &\hspace{-1cm}= 38 \, GGGGGGGG + 45 \, HHHHHHH,
\\[1ex]
    KKKKKKKKK &= LL \\
    &\hspace{-1cm}= MM + N.
\end{alignat}



ccc
\begin{align}
    AAAAA &= BBBBBB \\
    &= CCCCCC + DDDDDDD,
\end{align}
\begin{align}
    EEE &= FFFFFFF \\
    &= 38 \, GGGGGGGG + 45 \, HHHHHHH,
\end{align}
\begin{align}
    KKKKKKKKK &= LL \\
    &= MM + N.
\end{align}
\end{document}

Sorry, but you're the cause of your own troubles.

There are too many arbitrary added spaces, that make the formulas harder to read, both in the code and in the output. The use of \tensor can be avoided and the subscripts be vertically aligned nonetheless.

I define a \ts command that prints a phantom superscript dag, if not called as \ts* (that also recalls transposition).

By removing the added spacings, the formulas fit in the assigned text width. I also decided to align also the last equals signs in the formulas, so they're not lost by the reader at the end of a very long term.

The code:

\documentclass[11pt,letterpaper,twoside]{book}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[total={6in,10in},left=1.5in,top=0.5in,includehead,includefoot]{geometry}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{mathtools}
\usepackage{xparse}

\NewDocumentCommand{\ts}{sm}{%
 #2\IfBooleanTF{#1}{^{\dag}}{^{\vphantom{\dag}}}%
}

\begin{document}

Blabla bla bla bla blablabla bla blabla bla bla blablabla
bla bla bla blablabla bla blabla bla bla blablabla
bla bla bla blablabla bla blabla bla bla blablabla
\begin{align*}
\ts{\gamma}_{i} \ts{\gamma}_{j} + \ts{\gamma}_{j} \ts{\gamma}_{i} 
&= ( \ts{a}_{i} + \ts*{a}_i )( \ts{a}_{j} + \ts*{a}_j ) 
 +( \ts{a}_{j} + \ts*{a}_j )( \ts{a}_{i} + \ts*{a}_i )
\\
&= \ts{a}_{i} \ts{a}_{j} + \ts{a}_{i} \ts*{a}_j
  +\ts*{a}_i \ts{a}_{j} + \ts*{a}_i \ts*{a}_j
  +\ts{a}_{j} \ts{a}_{i} + \ts{a}_{j} \ts*{a}_i + \ts*{a}_j \ts{a}_{i} 
  +\ts*{a}_j \ts*{a}_i
\\
&= 2\ts{\delta}_{ij},
\\[1ex]
\ts{\gamma}_{n + i} \ts{\gamma}_{n + j} + \ts{\gamma}_{n + j} \ts{\gamma}_{n + i}
&=-( \ts{a}_{i} - \ts*{a}_i )( \ts{a}_{j} - \ts*{a}_j )
  -( \ts{a}_{j} - \ts*{a}_j )( \ts{a}_{i} - \ts*{a}_i )
\\
&=-\ts{a}_{i} \ts{a}_{j} + \ts{a}_{i} \ts*{a}_j + \ts*{a}_i \ts{a}_{j}
  -\ts*{a}_i \ts*{a}_j - \ts{a}_{j} \ts{a}_{i} + \ts{a}_{j} \ts*{a}_i
  +\ts*{a}_j \ts{a}_{i} - \ts*{a}_j \ts*{a}_i
\\
&= 2\ts{\delta}_{ij},
\\[1ex]
\ts{\gamma}_{n + i} \ts{\gamma}_{j} + \ts{\gamma}_{j} \ts{\gamma}_{n + i}
&=-i( \ts{a}_{i} - \ts*{a}_i )( \ts{a}_{j} + \ts*{a}_j )
  -i( \ts{a}_{j} + \ts*{a}_j )( \ts{a}_{i} - \ts*{a}_i )
\\
&=-i\ts{a}_{i} \ts{a}_{j} - i \ts{a}_{i} \ts*{a}_j
  +i\ts*{a}_i \ts{a}_{j} + i \ts{a}_i \ts*{a}_j
  -i\ts{a}_{j} \ts{a}_{i} + i \ts{a}_{j} \ts*{a}_i
  -i\ts*{a}_j \ts{a}_{i} + i \ts*{a}_j \ts*{a}_i
\\
&= 0.
\end{align*}

\end{document}

\documentclass[11pt,letterpaper,twoside]{book}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[total={6in,10in},left=1.5in,top=0.5in,includehead,includefoot]{geometry}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{mathtools}
\def\z#1{\llap{$\mathrlap{\displaystyle#1}$\hspace{3.5em}}}
\begin{document}

\begin{alignat}{2}
    AAAAA &= BBBBBB \\
    &\z{= CCCCCC + DDDDDDD,}
\\[1ex]
    EEE &= FFFFFFF \\
    &\z{= 38 \, GGGGGGGG + 45 \, HHHHHHH,}
\\[1ex]
    KKKKKKKKK &= LL \\
    &\z{= MM + N.}
\end{alignat}

\end{document}