Looking back in time by looking further away I have been pondering the question of how we can look back to the state of the early universe for some time. So I came here and found, to my delight, that several such questions have been asked, such as this one. Fortunately the answers to that (and other questions) has helped immeasurably, but I have one lingering area of confusion:

I often hear, in cosmological TV documentaries (such as Horizon in the UK) words such as "The further away we look, the further back in time we are looking. Our technology has now advanced to the point we can see objects so far away they began emitting their light just [n] after the big bang" [this is not a quote from a programme but my synthesis of what I have heard].

My question is this: I understand the limitation of c and light taking finite time to reach us from distant objects. Having now understood that inflation caused the early universe to "move" so far away we are only now receiving its light; How can today's technology evolve into tomorrow's technology and enable us to see "older" light as we can only see what is arriving at the Earth right now. I.e. if a time-traveller turned up today bearing a better telescope from the future, I should not be able to use it to see further away and therefore back in time, because it is dependent on the age of the light arriving at my location.

The only solution I can think of is that due to tricksy inflation rate effects, the light of successively further objects is arriving at, essentially, the same time (when looked at over a universal age). But that doesn't seem to make sense and even if it did how would we sort the older light from the newer light from the same object if it is all arriving at once (notwithstanding red shift, which I do (hopefully) understand)...
Before the advent of telescopes, we could only look back in time from a few years (for nearby stars) to a few thousand years (for the most distant stars visible to the unaided eye). In addition to this, a handful of galaxies are visible without binoculars, letting us look back a few million years.

The first telescopes allowed us to see much farther, like hundreds of megaparsec, so we could look back hundreds of millions of years. As technology improves and our telescopes become better, we are able to observe more and more distant objects, and thus look further and further back in time.

The farther away an object is, the less light we receive, and hence the bigger telescope and better detector we need in order to see it. So, being able to look further back in time than previously is not about the light from more distant objects eventually reaching us, but simply a matter of developing the tools necessary to see it.

However, even the most distant galaxy we will ever see (which are at earliest a few hundred million years after the Big Bang, i.e. roughly 13.5 billion years ("Gyr") ago), are not farther away than the cosmic microwave background, which was emitted 380,000 years after Big Bang, i.e. ~13.8 Gyr ago, and which we have been able to detect since the mid-1950's. If we ever manage to build an efficient neutrino detector, we may be able to look back to seconds after the Big Bang, when these particles decoupled from matter and began traveling freely through space.