Agree - $1000 or thereabouts would be a good price for one in good shape, they definitely go for less than most No. 1's. Completed auctions on gunbroker show one sold for $1200 but was relisted for the same price and now remains unsold (non-paying buyer, perhaps?), another with a scope sold for $1075 but a couple remain unsold at $965 and $1000.
I've been watching the auctions for No. 3's on gunbroker for a while, toying with getting one to send to Roger Renner to turn into one with the pewter forearm shown here:
Btw, this is an old pic, it costs more than $750 these days...
The only No. 3 .22 Hornet I owned was very accurate, as in 5-shot groups around 1/2" to 3/4"--using the same handloads used in my Ruger No. 1B. Would still own it, except for going through a .22 Hornet purge a few years ago, after realizing I was using my .17 Hornet a lot more....
truegunvalue.com lists the average No. 3 in .22 Hornet selling for $1,232.54.
Highest price was $2400+ and lowest was just over $880. The high priced one skewed the average up so I think $1000 as noted by others is pretty close to what they will cost in okay shape.