It isnt clear to me (sorry if I missed your posts) whether this is your family's own land and your horse or whether both belong to a RS.
I have hacked a lot including hacking solo on a share horse. What you need to remember is there are two creatures involved. The first is the horse. Does the horse know this field? Does the horse pay attention to you in the school? If the answer to both is Yes, then you should be fine to ride round the field.
But riding a horse out hacking is like taking a toddler for a walk - you are in charge.
Yet you need to pay attention to input from the horse. For instance, supposing the horse does know this field but is used to cantering across it - Horses learn their jobs and if the horse has been asked to canter in a particular place, it may be hard to stop him doing the same when you are riding him. In the UK riding schools the rule is that one must be safe in canter before hacking out and this is because you need to be able to stay on a horse if it runs away with you. I hacked before I could canter properly but that was because my teacher had seen that when a horse did unsolicited canter in the school, or spooked, I didnt fall off but took control.
I am totally in favour of hacking and learning to ride out hacking. I did it myself and still regard horses as a means of transport and a way of enjoying the landscape. But you have to understand that depending on the time of year and the weather and what the horse has been fed, its reactions can vary and this is especially true in wide open spaces. So if you are riding out, make sure you stay safe. I have a good hat and I wear a body protector too. And I still have to deal sometimes with unsolicited canter from an over excited horse - I have seen her do the same when ridden by an expert rider, a professional, so dont think it has anything to do with being a beginner.
Teachers and professional trainers fall off out hacking because hacking is at times and on some horses more complicated than riding in an arena. The first step is doing what your teacher suggested - riding the horse along paths close to the arena. And trail riding like you have. That is what I did too. But it is like anything one does with horses. Take it slowly, and follow the advice of your teacher.