I have given away two horses (both were dangerous). The first we advertised as free, with full disclosure of all of her problems, but did not include a name. We were contacted a couple hours later by a rescue who had owned her before the people we bought her from, who were aware of her issues (they were pain related, and because we had not been told by the previous owner, had not been able to manage). They took her, got her back on track, and now I believe she is used as a trail horse suitable for all ages.
The second horse I gave away as free was advertised for cheap and we still got all the wrong people asking about him (mostly young teenage girls who called themselves 'professional trainers'). We found his next home through a friend, and now he's happy as a clam outside of the arena, toting his new owner around to get his daily shopping done
Honestly, I think the BEST way to find a 'free' home for your horse is through word of mouth, people you know and who will give the best care. If you're unable to do that, then I suggest you just be very careful / selective of who you pick, look at multiple homes, visit the home or stable the horse will be kept at and ask for references (vet, farrier, trainer, friends, etc.)