As far as I remember you are a native English speaker so you know the difference, but just repeat, perhaps for others: "We also need that swift horse." would mean some particular that horse which is also swift, so that swift horse. We need that particular horse, which also happens to be swift.
We also need a horse that swift. means we want A horse that is that swift as something we probably can see or what somebody just told us about. Or as swift as the horse the speaker we are talking to owns.
Tak rychlého is literally so swift, so fast, so indeed we (také - also) want a horse that is so swift as something we know.
Ah, so, if you insist on using "that" rather than "so" or "such", it literally means "We also need that swift a horse." It is a fine distinction, but "We also need a horse that fast/swift" though generally misused literally means something along the lines of ours needs its oats (equine equivalent of Wheaties for breakfast) too . . .