I would like to know what しっかり頑張ってこい means and if it has any difference (any extra nuance) with がんばって.
Why is こい (command form of くる) used?
In which situation do you use this expression?
It is fine to say がんばって to anyone, but you can only use 頑張ってこい if you are higher in the honorific status (otherwise it will be rude). This is because 頑張ってこい is a command.
しっかり means "properly", and thus you should also only use it if you are higher on the honorific status. Both しっかり頑張って and しっかり頑張ってこい is possible. The former form is considered "softer". For example, women will tend to avoid しっかり頑張ってこい and instead prefer しっかり頑張って.
〜てこい is a command to perform something. For example, one can also say 洗ってこい、食べてこい、やってこい etc. The difference with 洗え、食べろ、やれ is that it implies you go somewhere to perform these things, rather than doing it here.
To the point of @ericfromabeno, I agree. If somebody can say 頑張ってこい (due to the honorific relationship), but chooses 頑張って, then they are expressing the wish for you to succeed. They won't use this form if they are insisting on your success. For example,
「三等兵、危険な任務だが頑張って」 ← odd 「三等兵、危険な任務だが頑張ってこい」 ← natural 「決勝戦絶対に負けるなよ、頑張って」 ← odd 「決勝戦絶対に負けるなよ、しっかり頑張って」 ← less odd 「勝ち負け気にせず頑張って」 ← natural
I think the main difference is pressure. If you don't want to give someone pressure, you use 頑張って. If you want to create a positive pressure you add しっかり or even ってこい. You can only create pressure without being rude if you are "higher" on the honorific status though.
There was a question just a couple days ago about 呼んでくる, and the meaning of くる in that case, which might also apply here.
Without knowing the context of the しっかり頑張ってこい it's hard to know for sure whether the こい is as you mentioned, an actual "command to come (back)" or whether it is くる being used in its "emotional" sense, representing a psychological "approach" to the speaker, or "growth/change".
The major semantic difference is that 頑張って(ください) is a simpler well wishing phrase that less emphatic, and doesn't make assumptions about the results of the other person's efforts.
頑張ってこい on the other hand, by including the "command" form of くる, can be thought of as more emphatic/assertive, more positive, possibly even predicting or insisting on the success of the other person.