Although I don´t claim expertise in modern approaches to EFL, since I qualified in it some 23 years ago, I teach Internationally in English for mainly non-native speakers and find that vocabulary based approaches often work better than the proscribed coursebooks. The relevance of the language plays an important part for the learner. It is no benefit to a Business User learning 3rd conditional without the basic vocab to speak to a potential customer. And I think that the different methods in teaching EFL over the past 20 years or so have not led to a clear winner in respect of success. Essentially the importance is in communication withing the target language and lessons should be geared towards this communicative ability along with being appropriate level of course.
Good academic work is coming out of Sophia University in Tokyo concerning the implications of plurilingualism, among other issues, for second languages education in Japan. In this regard, the work of Prof. Kensaku Yoshida is very thought provoking. Plurilingualism is a concept that encourages people to view themselves as users of language (s) rather than perpetual learners, a view that is in sync with the Multicompetence work of Vivian Cook. As a general concept, rather than a teaching manual, I strongly recommend this article for you, wherever you may be working. http://pweb.sophia.ac.jp/1974ky/Reconsidering%20Japan%81fs%20English%20Education%20Based%20on%20the%20Principles%20of%20Plurilingualism%20%28revised%29.pdf
David Coulson has recommended a feasible trend by introducing the concept of 'plurilingualism'. Having read the article by Yoshida, I must say that this concept could be the new wave in EFL. However, this is one that has to be strongly delegated from the Educational sector of the ruling Govts. in any non-native English speaking country. This is not without its own constraints. A long road ahead, but hope the 'Can-do' statements can be achieved through first and foremost a change in the mindset of the administrative elite. In the meantime, TBLT and PBL offer great opportunities for enhancing critical skills needed for the overall success of the lessons being taught and learned.
As far as I am concerned, the most famous approach nowadays in teaching language is the Communicative Language Teaching Approach (CLT). This approach is definitely promising. As a matter of fact, when people claim that they speak a language, they evaluate themselves based on their ability in conducting a conversation. That is, speaking is the most crucially important skill in language learning in this day and age. This approach helps language learners to learn faster and be better communicators of the language they are studying as they learn language based on communication. It is completely different from Grammar-Translation method as Grammar-Translation Method slows down the process of learning. It has to be mentioned that this approach is more student-centered learning. In a nutshell, I find CLT the most important approach in teaching a language.
As I see it the most recent approaches are the competency-based approach, the communicative approach, and the computer-mediated learning. All of which are learner-centred approaches
If you are interested in teaching grammar, the task-based approach which includes the PPP stages - Present, Practice, Produce is one of the latest teaching approaches.
If you mean the latest EFL approaches in a more general sense - CLIL (Content and Language Integrated Learning, which is a competence-based approach) is gaining popularity and includes teaching both EFL and the subject taught in English.
Blended Learning is another trendy approach - using the latest technology to teach EFL via digital and on-line media at student's own time and pace.
I reckon we have started moving beyond approaches and methods since a decade ago, and now its time to design, develop, and practice courses, materials, and tasks germane to learners' needs, considering environmental factors. I would highly recommend you Kumaravadivelu's book "Beyond the Methods" (2002), which introduces Micro and Macro strategies for language teaching.
Indeed, we are well beyond the method era, so an eclectic approach would be the desirable way of teaching a class. A little bit from here, a little bit from there, ... taking positive aspects of any of the traditional, alternative and modern approaches and methodologies to suit your students' needs.
In learning languages, a distinction is usually made between mother tongues, second languages, and foreign languages. These language learning theories are i.e. audio-lingual approach, the new natural-communicative approaches, the humanistic approaches, Communicative language teaching, Language immersion ,Community Language Learning etc and how they influence language teaching methodology today is called applied linguistics. These theories influenced by developments in the fields of linguistics and psychology.
I guess one must be more responsive to their students' learning styles than come up with a rigid methodology. Eclecticism is the less dogmatic way of responding to learner-centredness. Besides, one should never forget their students' background and their learning habits, in particular the way they have learned other languages (national & foreign).