Hi Abolfazl,
"doch noch" is a very common combination - especially when something is expected to happen, but turns out the opposite way, like:
"Trotz seines Sturzes kam der Radrennfahrer doch noch als Erster ins Ziel."
Meaning "Despite his fall the racing cyclist finished first."
"Um mit dieser Achterbahn zu fahren, bist Du doch noch zu klein."
Meaning "To ride this rollercoaster you are still too small."
Maybe parent and child measured the child´s height in the morning (1,25cm for example). But though they were expecting to be able riding a certain rollercoaster, at the fair they discovered it is not save for children under a height of 1,30cm.
In your case it is used - as you already guessed - to turn the imperative into a suggestion and also to set the events into a chronological order. A similar example would be:
"Wir gehen nach der Geburtstagsparty vielleicht noch in eine Bar oder einen Club. Komm doch auch noch mit!"
"After the birthday party maybe we´re going to go to a bar or club. Come on, join us!"
The first "noch" emphasizes that other locations are "added" to the evening (bar/club).
The word "doch" implicates that they are afraid you would otherwise go home if not invited.
The following "auch" includes you in a larger group of people, and the second "noch" repeats the chronological order of events.
It is more like "Let´s not seperate after the party already. Why don´t you join us for the rest of the evening?!"
So it doesn´t sound like a command/imperative and it "adds" another shared activity to the time already spend together.
Hope this explanation is of any help to you?