For work as an oral examiner, you could contact UCLES in Cambridge or their representative in Italy (you might get the name of this person through the British Council). You've already worked as an oral examiner for the Starter exam, you could try to do the same for other children's exams - Flyers, etc. You could also do oral examining for adults e.g. FCE and Proficiency. To do this work, you'll need to attend a meeting where you learn what's required and standardise your work with other examiners. With time, you might be able to lead these standardisation sessions yourself. You could also look at being an examiner for TOEFL, GMAT etc. Check their websites for information. You need to remember that most exams are held once or twice a year, so you won't get this sort of work often. Also when exam time comes around, you need to be available for full or half days. This might be difficult if you have other students, commitments, etc.
As for training work, you've done well in getting the Diploma. You could start looking at ways to train and develop teachers where you work to give you experience. You could offer to give professional development sessions on a subject of interest to your colleagues. You could also explore the possibilities of being a mentor to less experienced teachers. This involves being available for teachers when they have questions or concerns about their classes. This is one aspect of the work of a trainer. You could also look at carrying out observations and giving feedback to colleagues. Does a system of observations exist already in your school? If so, you can speak to the person in charge. If not, you can talk to the DOS about possibilities in this area. A system of peer observations might be useful. Take care with this as some teachers can feel uncomfortable or insecure about being observed.
You could also give a presentation at a conference. Check with organisations such as IATEFL and TESOL for dates of conferences near you. The benefits here are two-fold: you'll get experience of training and make useful contacts at the same time. At any presentation you give (whether at a conference or with colleagues), ask participants to give you feedback. A short, easy-to-complete questionnaire would be the easiest. This feedback will help you to develop as a trainer.
Finally, IH in Hastings used to do a course for teachers wanting to become trainers. You could check whether they still offer this course.