I would like to stress that the following is my own experience and personal opinion. That's how I did it - the same may or may not work for you.
First of all, the ~2000 jouyou kanji are not "all" the kanji (not even with the latest addition). When I take a list of jouyou kanji, there are some I'm surprised they're even in there, because there quite rare (=I haven't seen them often.) On the other hand, there are quite some non-jouyou kanji I encounter more often. I haven't kept track, but I would expect the number of non-jouyou kanjis I've seen to be >1000 (excluding proper names).
Now, it depends on what you mean by extensive reading, in particular what kind of texts you plan to read. There are many technical/cultural terms that use non-jouyou kanjis.
Expect to find many non-jouyou's in literature (including, to some extent, "short novels", "visual novels", "video games" etc.) and in old writing (~everything before 1945).
Say if in a year, I am reading something and I recognize the kanji but don't remember it, I can always look it up and it will jog my memory for next time whereas if after I learn 80 kanji and practice those 80 until I know them, I'm still limited to knowing those 80 kanji.
That is my personal experience as well. It makes quite the difference for me when I come across some kanji (or word) that, while I can't quite remember it, I am *familiar* with, as in I can remember having seen it already. Some kanii (or word) I've never seen before is another story, it feels, well, new and in a strange alien and strange to me. Learning something from scratch is harder than strenghtening your memory.
With said being said, if I were to learn the kanji again, I would proceed as follows: Study at most the ~1000 taught during the first 6 years of school via the method of your choice - it helps to learn a word (or sentence) that uses it as well. Start reading something of interest to you (eg a short story, newspaper articles (?), blogs &c., but perhaps not Japanese twitter &c. as you will find colloquialisms and abbreviations that make it hard to understand) - the point here is not being able to read everything already, consider it as an alternative way of study. Take note of the kanji you come across here and add them to the list for your favourite method of study. Doing it this way will teach you the kanji you actually need to know, convince you that you actually need to know them, provide context and words that show how to use the kanji and how they are read in compounds, supply you with a positive memory of those kanji (and not "oh that's boring kanji 17xx from that huge boring list that looks the same as #13xx and means the same as #19xx ... who cares); and finally you will get in some real reading practice.
Keep in mind this is my own personal experience and opinion. Try different methods, and if something works for you, it is a good method. If you feel you can learn all jouyou's quickly and with ease, feel free to do so. If you have a hard time learning the first 1000, learn less and start reading simple texts/textbook sentences/online lessons.
One medium that is, unexpectedly, pretty helpful -iff you are interested- are subtitled movies and certain kinds of (story-based) video games: You get the sentence written down, it won't go away until you press a button (or you can pause it) and you can take all the time you need to look up unknown kanji, it is accompanied by voice-acting recorded in a professional environment and gives you a crystal-clear pronunciation without background noise, provide you with a plethora of contextual information to aid your understanding (visual, auditory, coherence ie. what has happened until now, pre-knowledge ie when you have played/watched an English version already).
Don't forget grammar. There are some good guides out there (eg tae kim's guide, and nihongoresources). They provide you quite an extensive "overview" - you won't remember everything when you first read it, but the same principle as above applies - you will be familiar with most of grammar that stands in your way of reading comprehension and you can always look it up and strenghten your memory thus.