Since your rabbit prefers the coarser pieces, I would recommend that wherever you choose to buy 1st cut if you can find a place that differentiates between the cuts. 2nd cut timothy does have coarse pieces, but from my bale, it’s probably about 50/50 coarse vs soft.
from rabbit.org:
“First Cutting: The first growth off of a field for the year is the “first cutting.” Many people erroneously feel that first cutting hay is not to be considered as good feed. We tend to disagree, provided it is of good quality and was cut when relatively immature (pre-bloom stage), before the plant is allowed to mature to the point where the stem becomes larger and coarser. This is when the lignin (an indigestible part of the fiber component associated with cellulose and hemicellulose in the cell wall) content has become sufficiently high so as to make the hay more unpalatable and indigestible and the nutritive value has declined greatly. This can happen with 1st, 2nd, or any cutting of hay if left growing too long.
Second Cutting: Depending upon the temperatures of the days and nights, it typically takes 40-45 days for regrowth of alfalfa, mix hay, and orchardgrass , and 55- 60 days for regrowth of timothy. This is termed the “second cutting,” which usually has a larger percentage of leaves to stems, has a finer and softer stem, has increased percentages of crude protein and crude fat, and has a lower crude fiber percentage (depending upon the stage of maturity at which it was cut) . More non-structural carbohydrates (starches and sugars) and protein are in the leaves than in the stems. These starches and sugars are very digestible and make the hay higher quality.
Third Cutting: If the growing season is long enough on any given year, it may be possible to secure a third cutting. In regions that lie south of our location, the growing season is longer and hotter, making alfalfa the prime hay crop, and often as many as four or five cuttings may be taken from a single field.
The third cutting is typically very soft hay that is primarily leaves with very few small stems. While beautiful to look at, it can be “rich” (high in nutrients, having a high Relative Feed Value or RFV, and low in fiber). It is our opinion that third cutting hay does not contain sufficient fiber content to be the only hay in the diet of most rabbits. It can, however, be used in conjunction with a higher fiber, good quality, relatively immature 1st or 2nd cutting hay, and creates greater variety and interest in the chewing experience. We suggest that you feed the different hays at different meals so as to eliminate waste.
FORAGE QUALITY
The stage of maturity at which forages are cut (whether it be 1st, 2nd, or 3rd) has a major influence on the quality of that forage. Forage crops generally decline in nutritive value as they mature. As forage plants mature, it is typical for an increase in Acid Detergent Fiber or ADF to occur.”