Most serious adult athletes will not experience any improvement in VO2max (absent weight loss or a move to altitude). If they improve at the 3k distance (roughly vVO2max), it's because of increased economy (which increases velocity at the same oxygen consumption level) or anaerobic tolerance.
LT, on the other hand, can continue to improve for years and get closer to your VO2max. That said, LT improvement will also level off in the vicinity of 90% of vVO2max, and it only takes a few years to get there as well, depending on how hard you train. To get LT faster after that, you would need to improve economy (the only way to increase vV02 since VO2 is probably fixed). That can potentially be done with drills, plyos, hills, etc. Depending on your current ground contact time, there may not be a lot of room for improvement there. It's also the natural result of lots of running at a variety of faster paces, which you're probably already doing.
Keep in mind that when I say that these values level off, I'm talking about long term development. Both will degrade quickly with a layoff from running, but you can get back to your previous level pretty quickly with training.
Now, as others have pointed out, there's nothing magical about LT. Scientists and coaches get worked up about it: (1) because it correlates with performance better than VO2max, (2) because it responds to training over a longer period, and (3) (here's the big one) because it can be measured in a lab. (Though even most lab step tests are really just good guesses.) The thing is, all three criteria are also met by longer distance races (10k to half marathon). They correlate strongly with performance at other distance, they're more trainable than shorter distances, and it can be measured precisely with a hard race. Scientists will quibble that the actual race involves psychological factors that are absent in lab testing of LT, but I think the margin of error for lab testing is significant enough that we can call that a wash.
So what do we do with our longer distance race pace? We compare it to our 3k race pace to determine the Riegel Coefficient. Basically, there's a fairly predictable rate of decay in performance as distance increases. An 8% slowdown in speed as distance doubles is a standard rule of thumb. But well trained athletes can get down to 5% or POSSIBLY a little lower if they have an extraordinarily high percentage of slow twitch fibers. So if your Riegel coefficient is higher than 6, then you definitely have room to bring your longer distance PRs up to the ceiling that's set by your 3k PR. You can calculate your personal Riegel Coefficient using this calculator:
https://www.runningahead.com/tools/calculators/race
. Plug in two races. Ideally the short one would be 3k, but 5k is okay. Plug in different distances for the second one and see how the coefficient changes. It's not unusual for the coefficient to be pretty low at 10k but to drop off at 10 miles to the marathon. It's good news if it drops off because those are races where you probably have untapped potential.
So turning to your particular situation:
1) If you're improving at distances between 5k and the half marathon, your LT is almost certainly improving.
2) It's important to be realistic about the improvement that you're going to see. An experienced runner is not going to see massive improvements in LT. Five seconds per mile would be a very big improvement, and it's not necessarily something you're going to "feel" on a normal run.
3) If your Riegel Coefficient for your 3k (or two mile) PR and a race around one hour is over 6%, then you have room for improvement still.
IF the third situation applies, then you need longer workouts at paces ranging from 10k to marathon. There's nothing wrong with straight LT runs, but that shouldn't be all you do, or even your main training tool. When I say long, I mean stuff like 9 x mile at 10k pace or long runs alternating LT miles with just slower than marathon pace. There have also been some good suggested workouts on this thread. In layman's terms, you need workouts that test your legs rather than your lungs.