Versailles Garden Entrance

Are you taking the train to Versailles or driving? If you have a car you can drive onto the grounds at the north edge of the gardens through the Queens Gate and park right near the Petite Hameau and the Trianons. There is another lot in the trees right next to the Grand Canal to the back of the chateau - you can walk up to it from the rear. Here is a map of all the lots on the grounds. It's cheaper to park in these than in the main lot at the front of the chateau.
I haven't gone into Versailles through the front entrance in over 15 years so I'll let someone else address coming in from that direction...you used to be able to walk right through into the gardens but that may have changed.

When we were there 7 wks ago you could walk right into the gardens from the front. No going through security for the gardens. Security was inside the palace.

We did exactly this in late June of last year. it is a first-rate beat-the-crowds strategy.

There is a city bus that will take you form the train station to a stop that is not far from the Petite Trianon, which is at the other side of the estate from the palace.

You probably know that if you enter the gardens near the palace, you will have a good deal of extra walking to do if you are going to take in the Hameau and then return. Although there are transit options within the estate: bicycle rental and an electric tourist "train."

Have a great time!

Note that the Trianon, including Marie-Antoinette's Hamlet (my wife's favorite part of the whole place) doesn't open -- at least currently -- until 12:00 noon. (I think we took more photos of the critters and buildings in the Hamlet than we did of the Palace interior.) Your plan sounds great.

Our favorite way to see the grounds, the two Trianons and the Hamlet is by bike. Very fun. Rentals are by the Grand Canal.

The entrance to the gardens when approaching the chateau is around to the right whereas the big security line for the chateau is out the front. When the fountains are not running, there is no line or security into the gardens. You just walk in the gate there. There are a couple of other entrances, one at the bottom of the gardens and one off to the side if you are entering from the estate.

If the fountains are running tickets are required and there are ticket booths there. We went in October of 2015 on a fountain Sunday and there were maybe 6 people in line ahead of us to buy garden tickets. So there won't be a big security line regardless of when you you if the gardens are your object.
https://janettravels.wordpress.com/2016/02/15/fountains-of-versailles-a-little-chaos/

Every time I've been to Versailles, 30+ times, for decades, and 7 wks ago, the front entrance to the gardens was on the left side of the palace when facing the front of the palace, not to the right.
And security was inside the palace 7 wks ago, as I said above. That could change though.