It took me nearly 3 yrs before I could touch my 3rd SV who is a F2 male. He came home at 14 wks of age with giardia and coccidiosis and I had to wrap him in a towel to give him his meds. He had been weaned the week before I picked him up. On top of that, he wasn't what I consider properly socialized by his breeder. And to make matters worse, he is the most sensitive animal I've ever met. He gets worried if we raise our voices in laughter.
I tried everything to get him to trust me, never forced myself on him after he was finished with his meds. The biggest thing that turned the tide in my favor was when he was introduced to my other 2 SVs and could watch me interact with them and pet them. At that point, I could touch him when he was resting but he would flinch and seemed to tolerate rather than enjoy the interaction. i complicated the matter, i'm sure, by not being in a proper frame of mind when I tried to pet him. I was scared he'd never accept me the way I wanted, he picked up on that and it made him uncomfortable

So I made sure to relax and get in a positive frame of mind before I tried to touch him.
To make a very long story less boring, I kept touching him when he was resting or sleeping or any other time he got close enough -- just a short stroke or two and I gradually increased the amount of time I'd touch him. Sometimes I'd just hold his paw or his tail when he sat beside me. He's 3 yrs old now and we've made steady progress. He'll even purr for me on occasion.
Your F1 may or may not tolerate touch. If his body language doesn't indicate otherwise, perhaps he communicates by growling. My problem child is a hisser as was his mother. He hisses more often than he meows. I misinterpreted those hisses as displeasure and would back away. He learned to use that to his advantage. Then I just got fed up with the whole thing and decided I was tired of letting him call the shots. When he'd hiss, I'd talk softly to him, pet him with regular strokes and when he realized I wasn't going away, he'd stretch out and enjoy it. If he appeared as though he would bite or scratch me, I'd stop touching him but continue talking to him. I can now play with his ears, pet his belly, scratch his chin but only when he is resting. I cannot pick him up. He can't be handled at the vet's and has to be anesthetized for an exam.
Be patient. A month is not a long time for a cat, especially an older one, to adjust. You just have to keep trying, be gentle but firm, and don't make the mistake I did of letting negative emotion prevent progress. In the end, my Maliik is a cats' cat. He prefers the company of the other SVs to humans. We can only do our best to establish trust and a bond but in the end, we have to respect the limits set by the cat.