array_push() with key value pair

I have an existing array to which I want to add a value.

I’m trying to achieve that using array_push() to no avail.

Below is my code:

$data = array(
    "dog" => "cat"
);

array_push($data['cat'], 'wagon');

What I want to achieve is to add cat as a key to the $data array with wagon as value so as to access it as in the snippet below:

echo $data['cat']; // the expected output is: wagon

How can I achieve that?