I don't know how the used market is in your area, but most places around here are asking close to the same price you would pay for new - sometimes even more. For a .38 revolver under $300, all I'm seeing is the EAA Windicator. Bud's has them in 2" and a 4", both for under $300. I don't know much about them, so I'd recommend a bit of research before buying, but it seems to meet the criteria.
I actually had this limitation when I went to a gun show a couple weekends ago. (The misses gave me this haha). I looked around quite a bit and I found this.
Taurus M85 for 270 dollars. I know there are others, especially on the Taurus forums that have gotten them cheaper, But even at the academy here they are 300. So for a bit cheaper I got myself a nice little wheel gun that I think feels better than the SW Airweight my wife and I shot at the range. They even had a 605 at the show but it was 320 and it was a little more scuffed than I cared.
$300 won't get you far, so I"d opt to save up & buy something better than what 300 will get you. As to the used Smith Model 10's...and I really do like Smiths...it's a great gun in need of decent sights. Save your shekels and get a M19 for versatitlity or a M15 with adjustable sights. Trying to get most any fixed sigthted Smith of my experience to shoot to the sights is a Royal PITA.
If all you intend for the gun is sitting in a bedside drawer, the Smith M10 is ok as the distances are well under 10 yards. But for a more useful range of uses, adjustable sights make the most sense. Home defense at 'across the bedroom' distance...or using the same gun while thinning the jack rabbit population out to 50+ yds demands better sighting arrangements.
Just put $100 bucks down on a Armscor M206 spurless hammer, in nickel NIB with the wood and the rubber grips included. Total price new is $339 before tax at a local shop.
Just reiterating what I said about MSRP.
A while back I found a really nice former law enforcement S&W model 15. I pointed wife unit to it, and it became her bedside revolver. It was a bit less than $200 at the time. I have no idea what it would be now, but given inflation you might be able to find something similar for around $300 or so. Shop around, find a nice used K frame. Don't be scared off by a little holster wear.