How to tackle regular expression to e-NFA conversion with multiple Kleene stars?

Question

I need to clear up some doubts when it comes to applying Kleene closure (star) when converting re to e-NFA. I constructed the following regular expression: $(a+(a+b)^∗)^∗$, but what is bothering me is the second Kleene star. For the first Kleene star i.e (a+b)*, I e-transitioned from the final states (let’s call this Fin_1) to the old initial state (let’s call this Int_1), but now when I applied the second Kleene star to $(a+(a+b)^∗)$ there are new final states (let’s call this Fin_2) that will e-transition to the new initial state (let’s call this Int_2). So the question is, do I need to e-transition again from the old final states (Fin_1) to the new initial state (Int_2) or I don’t touch Fin_1 and just focus on Fin_2? I attached the following images to show what I mean: 

$(a+(a+b)^∗)$:

 

 

 

 

 

 

 

 

 

 

 

 

 

$(a+(a+b)^∗)^∗$:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Is the second graph correct for my regular expression or do I need to e-transition again from the old final states to the new? 

Comments

let’s your RE = (c+(a+b)*)*

By your second graph, cccca is acceptable. But ccccac is acceptable ?

I mean After entering into (a+b)* part, you’re not able to return ‘c’ part.

 

You need to observe, whether I need to get back or not.

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I see. Thanks.