Your prediction of panic caused by the contact is based on peaceful state of humanity. That is the panic can be amplified because this is going to be the biggest news, humans will pay attention to it and will take actions based on what they learned.
Make sure to check out the book "A.D. After Disclosure"
by Dolan and Zabel. This book is a bit biased because it is catered to mainstream public. Still you will learn new insights how the Disclosure might unravel.
Now consider making the disclosure of your existence as a piggyback action followed after a big event. Big events happen quite frequently on Earth, unfortunately. These are typically wars, terrorist acts and natural disasters.
If the contact is made in the shadow of a big event, it will have little or no effect on the news. People will be distracted from it. Only afterwards they will realize that aliens are here and there is no way to ignore this. The panic will be smaller.
The problem with piggyback disclosure is that MIC will
1. attempt to hide it or
2. to distort the news in negative sense for you. You will be presented as negative aliens and MIC will try to create panic at their initiative and timing.
These problems could be solved; I will not discuss the solutions here.
A GOOD ENTRANCE A good entrance is going to be for you to help with some rescue works in a natural disaster. Of course you don't want to have scarily looking aliens to be helping desperate people, but some things can be done in a non-scary fashion.
MIC = military-industrial complex