In all situations, you need to start by ensuring that you have spelled everything correctly in any written materials submitted.
Your need to contact consultancies is doubtful but might be influenced by the type of job you are seeking. If you are searching for an ESL job overseas, there are organizations to help make matches and arrangements between schools and teachers.
Before making any contacts, you need to determine the type of situation you hope to find. Where do you hope to teach? What age of students do you hope to have? What subject(s) will you be qualified to teach? Even if your answers are "Anywhere, any age, all subject areas", having some general direction before you start your search will give you some focus and a basis for further decisions.