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Is there a way to change operator precedence for built-in symbols?

I'd like \[TensorProduct] (which I redefine to KroneckerProduct) to take precedence after dot, so that the following is true

A.B\[TensorProduct]C.D == (A.B)\[TensorProduct](C.D)
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    $\begingroup$ Related: 140609. People tend to suggest workarounds rather than actual ways to change operator precedence. Edit You might give TensorProduct a HoldAll attribute and define downvalues (to TensorProduct) when a Dot object appears in TensorProduct. $\endgroup$ – jjc385 Apr 19 '17 at 16:24
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    $\begingroup$ Also, how exactly do you redefine \[TensorProduct] to KroneckerProduct? $\endgroup$ – jjc385 Apr 19 '17 at 16:35
  • $\begingroup$ I do TensorProduct=KroneckerProduct $\endgroup$ – Yaroslav Bulatov Apr 19 '17 at 20:14
  • $\begingroup$ How about using CircleTimes = KroneckerProduct and then A.B\[CircleTimes]C.D $\endgroup$ – chuy Apr 20 '17 at 16:46
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If you enter \[TensorProduct] using the input alias t*, then you can override the input alias to produce a TemplateBox with a SyntaxForm setting to lower the precedence:

CurrentValue[EvaluationNotebook[], {InputAliases,"t*"}]=TemplateBox[
    {},
    "TensorProduct",
    DisplayFunction->("\[TensorProduct]"&),
    InterpretationFunction:>(Sequence["~","TensorProduct","~"]&),
    SyntaxForm->"*"
];

Your example (as an image to show the alias in action):

enter image description here

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