Many people perceive the 1934 pattern field caps as being early war items, but they continued to be made and issued past the mid-point of the war. The cap on the top (or in the left) in these photos was made in 1942; the cap on the bottom (or right) was reworked for reissue in 1943.
The 1942 cap here is made from typical wartime wool with a rayon lining that is more commonly associated with 1942 or 1943 pattern caps. It has 1940 pattern insignia. The 1943-dated cap was originally an earlier cap that was reworked with new lining panels. Many other examples of this type of reworked M34 cap with replaced lining parts are known, some from the same factory. Usually these have the 1940 pattern insignia, sometimes they have been resized and still have original manufacturer information and dates in the upper part of the lining. This one has a mismatched 1937 pattern cockade with a 1940 pattern eagle. The factory that made this did make M43 caps as well; it’s not that they were a manufacturer who did not change over to the new pattern, rather they were contracted to rework these hats for reissue.
Both of these caps originally had branch soutache, removed after mid-1942. They have machine sewn cockades and hand sewn eagles. Many collectors might regard these eagles as replaced, especially the 1942 example which has terrible stitching. I think both of these eagles are probably original factory applications, and I have my reasons for thinking this, though there is no way to prove it.
There are a lot of differences between these two caps, including the location of the insignia and the height of the front flap. Even on these simple caps with relatively few features, pattern differences are immediately obvious.
There are persistent collector legends about late war made M34 caps with 1944 dates or with factory applied 1943 pattern trapezoid insignia. I have never seen one and if they existed they must not have been common.
Some have asserted that after 1943, the M34 cap was mostly issued to non-combat units, static units or foreign volunteers, etc. This could be true; these units had a low supply priority and got lots of reissued stuff in general. The reworked caps show the effort that went in to keeping these caps in the supply chain. Usable caps were reissued, anything that could be made serviceable was reworked for issue.
In any case, the M34 cap was still evident in huge numbers in 1945, and almost certainly was still being issued (if not actually manufactured) until the last days of the war. Some 1945 film footage of surrendering units shows the M34 cap still predominant.
There may be specific units that can be researched and proven to have predominantly had M43 caps by the end of the war. Beyond that, though, I don’t think an M34 cap in 1944/45 can be said to be wrong. I believe they remained more common than many realize.
This last photo shows 5 original factory made enlisted issue M34 caps, just to give an idea of the range of variation here. Look at the differences in the scalloped front edges, for example. None of these caps have the same ventilation eyelets.