Sunshine is amazing!

Here is the tablecloth two hours after being set out to dry:

I've folded the side that wasn't up to the sun on top of the side that was. You can see a distinct difference. Oh and the smell? What smell? It is gone.
I've turned the tablecloth over and set the timer for another hour.  I don't think it is going to take the rusty spots out completely, but I'll wait and see.  If it doesn't, I'll do the lemon juice and salt paste method. Then wash it out and dry it in the sun again.  I'm glad it is daylight savings and I still have a few more hours of sunshine.

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I may have gotten a little carried away.  I thought,"Hey this worked so well for the tablecloth, I should try it for all my old handmade linens."  I have quite a few doilies made by my great-aunt, or great-grandmother.  So I put them all through the same routine as the tablecloth.
The before pictures:

They are all next to a machine made doily that is actually white.

Here they are soaking in Woolite:

A rinse in the tub:

And here they lay out on the grass:

Only problem, the sun went and hid behind some clouds.  So they didn't get as bleached as tablecloth.  I am planning on doing this all again this weekend.  There are still some spots that need to come out on all of them.  But now I know that this definitely helps get them whiter, while being much gentler on the fabric than say chlorine bleach.