There’s nothing in the big bang theory that states the universe was tiny in size. It’s just that the current known dimensions were contained in a tiny space. The universe could have been much larger, even infinite in size when the big bang occurred.
The question is one of the curvature of the universe. If the universe isn’t infinite in size, it has to curve back upon itself. Therefore, it might be possible that we could observe the same part of the universe while looking in different locations. Imagine a telescope seeing a particular spot in the universe to the west, and then the same spot in the east. There are people looking at the background cosmic radiation looking for such spots.
By the way, there’s also a question on how constant are universal constants. It has been stated that if some of these universal constants were a bit different, matter as we know it couldn’t form which brings us to the problem of the just right universe. To get around this issue, some cosmologists have speculated on parallel universes with different sets of these constants, and we just happen to be in the one with the right constants because if we weren’t, we wouldn’t be here.
However, if the Universe is infinite, it could be possible that various parts of the universe have different sets of constants. Imagine if these values could vary over vast distances in an infinite space – say over tens of billions or even 100s of billions of light years. It could be possible that we live in a part of the universe with the right constants while most of the universe has the wrong values. There has been recent speculatin on the fine structure constant alpha, and whether we actually see differences between one side of the universe to the other.