Yeah, I have a starting clamp and use it on the outside of the frame on the first cross and pull tension on the second cross.
I like the option of using your extra floating clamp in the manner, but there will be an issue as most floating clamps have a thicker and thinner one. So your thicker one will clamp cross 1 and 2 together, then do another 3-5 crosses using the same clamp for the strings.
Once you get too close to the middle that wider clamp will be too big, so go back to the first cross and pull tension and clamp 1 and 2 with that same clamp and tie off that first cross.
Any way you do it though, you will have a slight issue, as that thinner clamp that was your starting clamp will be very small for clamping cross 1 and 2. I am sure it can be done, but it will be a bit of a hassle. Just invest in a starting clamp......or use a small vise grip from HD or Lowes on the outside, it will be much easier that way.