My ex racer used to weave badly. I understand that he is fairly new to you? (apols if I have confused you with someone else!
) If so, then the weaving may decreased as he settles, gets used to you and starts to realise that this place is "home" and "safe".
To begin with, my horse weaved so badly, I thought he would fall over. He was incredibly stressed by the move and the best thing I did for him was to ignore it. I don't like anti-weave grills, myself - if a horse is stressed, it is stressed and stopping it weaving and "releasing" the stress will only make it more stressed. My horse came from a private field with one other horse into a large, busy barn. The sheet metal roof on the barn also made a horrible noise when it rained and for a long time, he would completely stress out whenever it rained. He just didn't like the noise, the people bustling around and all the new horses.
The first thing I tried was a stable mirror, which helped a lot initially, although after six months or so, he ignored it. I also tried hanging distractions in the stable with him - hanging veg, boredom breakers, Likkits, you name it, I probably tried it. The best toy that worked was a feed ball.
I also discovered that after a while, it seemed to be a bid for attention rather than stress. A lot of people were going and trying to "comfort" him when he weaved. One person even blocked his weaving with their body until he clunked them on the head with his jaw!
I had to tell people to please just ignore him. If he got no attention for weaving, he would eventually stop and go and eat his hay. He seemed to find it more stressful to have people waving arms at him or trying to hang on to his head or patting him and trying to calm him down than just being left to get it out of his system for five minutes.
I definitely think turn out is the way to go. I bought my horse in the middle of winter turn out, which meant I had no choice about stabling on a night, but I went to fetch him as late as I possibly could and turned out as early as I could manage. When he was in, I would drop in his feedball, and by the time that was empty the barn would have started to settle and quieten down for the night, so he could relax and wasn't so uptight. Then it was summer turn out and he was outside 24/7. When I brought him back in for winter turn out the following year, the weaving had virtually stopped.
He does still weave from time to time, if he is worked up or upset about something (such as being on boxrest - in this situation he is already either ill or injured and more prone to stressing out). He bobs his head when he wants feeding, which isn't really "true" weaving", but the first time that he stood quietly when it poured down was a true acomplishment for us.
He will always be a weaver, but fortunately, over time, his weaving isn't really a problem any more. I've had his joints and feet checked out, and there is no excessive wear, so I just accept that his weaving is a bit like a smoking addiction. If he's stressed out or worried about something, it helps relieve a little bit of it, and it's his way of communicating with me that something is bothering him. When he is turned out full time, he doesn't do it at all.
Hope there's a few ideas here to help you tackle the root of his weaving. Good luck.