I love riding in Yellowstone, but I don't go up that way until late July or August,,  Mosty to avoid the bugs and let the spring thaw melt off and the river crossing to subside a to reasonible level. I'll give some suggestion, But they may not be the best choice for June, since I've never ridden the are that month.

In the Hayden Valley. There are numerous pull offs along the hiway, We have parked in those pull offs and unloaded our horses and ridden in from the trail heads located near the pull offs. You will mostly see Buffalo in this area. We occassionally see a bear or wolf pack. Both of which usually flee us as soon as they catch our scent.  The buffalo poise a greater threat, If you get to near them, They will charge. They act so calm and uncaring, It is easy to approach too close and find them, quickly becoming upset and charging.

There is a trail from Asay creek that goes over the mountain to Old Faithfull.  We ride about 1/2 way across that trail to Mary Mountain and Lake Mary. Have lunch and return. It's a great way to see some of the interior of the park and probably not see another person.

The trail is easy to follow,  It's is pretty level with just rolling hills

At the higher elevations of the trail, You enter into the forest, But again the it would be hard to loose the trail

Lake Mary

There is lots of water along the trail for the horses to drink. In fact in a few places the springs almost become boggy, Another reason I wait till later in the summer, so that they dry up some.

We also ride a lot around the South Entrance to the Park. The Two Oceans trail heads out from Jackson Lake Lodge and you can circle past Emma Matilda Lake and around Two Oceans Lake.

Again this are very easy trails to follow, But you will see a few hikers along them. And the outfitter at jackson Lake Lodge does take paying clients out this same trail.

This the view from Grand View looking back at Two Oceans Lake, The trail goes all the way around that lake.

From Grand View, You can look back across to Jackson Lake and the Grand Tetons

Another Favorite area of ours is riding in the South west Corner of the park.  There are numerous trails that access that area. You can primative camp along the road that skirts the park boundary and they just ride into the park on your horses.  We often start from the Bechler Ranger station and ride into Union Falls or up the Bechler River.

Again, Very well used trails that are easy to follow. The wild flowers are absolutely stunning in Late July ( another reason we go at that time vs earlier)

You do have to cross several rivers and high spring run off can intimidate those riders not used to river crossings. But this years mild winter should see the water levels lower much earlier in the summer. This poor gals horse tripped and dumped her in the river. And you can see we are all wearing jackets in Late July.

Again this is all primative camping, Just pull off in the trees and put up a highline or tie the horses to the side of the trailer

Yellowstone is very protective of their park.  You will be reguired to have current Coggins and travel papers for your horses.  The trailer will need to be swept clean before entering the park. No accumulated manure or bedding on the floor.  They don't want you bring hay into the park at all. So you are better off bringing pellets if you need to carry feed into the park.  The National Forest around the park will allow Certified Weed Free Hay. Another reason we camp out side and just day ride into the park.

There is also a wonderful trail that goes back in along Slough Creek over in the Lamar Valley.  It is a well used trail that often see flyfisherman hiking back in 7 miles to the upper meadows to fish. Plenty of room to park at the trail head.

Most of these trails that I have described are about 20 miles. Some are in 10 and return 10, Others are loops that again are about 20-23 miles in length

Just outside the park are too many trails to count.

East of Moran is Turpin Meadows and the trail head for Hawks Rest.  This trail takes you in to Hawks Rest. The most remote spot in the lower 48 states. It's an easy ride following the Thorofar trail. Turpin Meadows has numerous corrals, pit toilets, water hydrants and lots of parking. It is a heavily used trailhead and you will see lots of individuals as well as guides departing from that location

We often camp along the Hoback River south of Jackson Hole. Up Granite Creek or Cliff Creek. Again this is all primative camping, Just pull off under the trees and set up a highline

Granite Creek

The area around Jackso is a lot steeper than other areas of Yellowstone, But the trails are well used and not hard to follows

And the trails often stay right on the floor of the canyons in the area that the rivers have flooded in past years

Lots of water for your horses

and we often just hobble to let the horses graze

It's easy, just pull off the road and find a flat spot under the trees. I usually look for camp spots near streams or lakes so I can water the horses and not have to haul horse water.

Cook on a campfire and be stingy with your water. If you have it use your generator for what power you need.

You will rarely ever need A/C at any of these locations, More than likely, you will turn on the furnace, especially at night.

Everytime I see the title for this thread, I suspect that I've mis-interpreted the OP original request. I've detailed camping and riding your own horses, And I keep thinking the original request was for  renting horses for a single day ride. 

I admit, That I have total ignored the professional outfitters in that area. Since I bring my own horses.  I have seen outfit strings tied and waiting for customers at Jackson Lake where the trail head starts for Two Oceans trail head,  The 7 mile long gravel road into Turpin Meadows has one or more ranches with signs advertizing Trail Rides which would take you toward Hawks Rest and the South East corner of the park.  There is a pack string near Canyon Junction that I often see heading out as I drive past and their rides would be into the Hayden Valley area of the park. Another string I frequently see around the Rescue Creek area close to Mamouth Which would ride the Northern part of the park.  These all operate inside the park, So they must be Concessionaires licensed by the park service. So the Park Service should be able to provide contact info for them.

Outside the park there are outfitters that will trailer their horses to trail heads near the park boundaries and ride the Forest Service trails that surround the Park. These could be very numerious and scattered in every town close to the boundaries of the park. They would be licensed by the state and probably have some type of a Forest Service Permit to offer a "for profit" business that utilizes the Forest Service land for their business.

To the original question, I don't think it would really matter,Which area you should choose. Any of it would be a wonder area to ride and especially for folks coming for other parts of the country, Offer a ride that would not be forgotten.