Have hunted and camped the NW corner of the state many times. While a lot of places are leased, it is still somewhat frowned on by some landowners out there. But that mindset is no longer the majority any more, as property taxes in this state are crazy and every little bit of income helps. It really is going to depend on you as much as anything and your ability to relate to the people you are asking to hunt. I have never had a problem finding a place to hunt antelope, rifle or bow here.
The grasslands are public and do hold hunt-able numbers of antelope. Not Wyoming type numbers, but its doable. If hunting access is an uncertainty, then this is where I would go. That way if you do not find private access, you can always fall back to the grasslands. I have seen big bucks out on the grasslands and we have got a few nice bucks off there over the years. But, far as Nebraska antelope goes though, the best hunting is on private land. The nice thing about Nebraska is most all of the leases are deer hunters and antelope are an afterthought. Still, leased is leased and you won't get in. What you don't have though, are a lot of outfitters trying to get every inch of land for antelope. The best place I have to hunt antelope is nothing special for deer and most deer hunters would never look twice at it. BUT is has a working windmill, which is golden far as I'm concerned. There are these kind of places, but you are going to have to knock on a lot of doors. It took me 4 trips out west that summer to find this place. Some in Crawford got to know me pretty good and eventually it paid off.
Far as other areas to hunt, I have or know people that have hunted in Banner North, Banner South, Box Butte West Box Butte East and Cherry. The all have antelope and some nice bucks, but numbers are less in these areas. A few are better than others with regards to numbers and trophy potential. One of these areas is my favorite for trophy antelope! Access is also more difficult, as some of these are great deer hunting areas and as PDR pointed out, there are lots of leases. The national forest at Halsey (Cherry) use to allow archery antelope and I was with a guy that arrowed a nice buck there years ago, but I do not know if they still allow antelope hunting there or not these days. The numbers are low, but the are a few there. You will see people, as you will have campers, people scouting for rifle deer season and a few archery deer hunters. Personally I would pass on that place.
The picture in my avatar is of a Nebraska antelope my son took a couple years ago. It was on private property that butts up to the grasslands. What happens on the grasslands, is a lot of the antelope hang out on these private lands next to the grasslands. Some of it is pressure, but some is also due to consistent water that is needed for cattle, as well as some alfalfa crops that are around.
Hope this helps,