Why も in 私も暇な時、犬といっしょに・・・


Can somebody explain も here to me please? And why it is not の?

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    私も暇な時、犬といっしょに近所を散歩します
    
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I know it is used to mean with or also (私もです - me too), but the above to me
would be 私の暇な時 showing possession. Or is も meaning "when" somehow?




Parse it this way:

> 私も｛暇な時、｝犬といっしょに近所を散歩します。

私 is the subject of the main clause 犬といっしょに近所を散歩します.  
暇な時 is a subordinate clause.

"I also take a walk in my neighborhood with my dog {when I have free time}."

The も means "also/too", as in "I take a walk too, just as someone else does."

