I use second crop (that my bunnies love) everytime one of my rabbits seems to eat less hay. A lot of people also give orchard hay or, in France, the hay for privileged rabbits is “Crau hay” (it’s the first ‘animal food’ to get a PDO, it’s used in some fancy restaurants for daring recipes for human too – it’s pretty pricey is what I’m trying to say ^^). So, you don’t have to use timothy, it’s just the more commonly used because it’s generally cheap and just fine for rabbits. To my knowledge, you can use the hay you want except for alfalfa (too rich in calcium which might cause problems later on like kidney stones). The only problem which might occur with second crop or crau is that there are flowers and nice smelling plants in them and some rabbits might pick them and leave the necessary long strands out. Mine don’t, if they like one hay bag, they will eat most of it (and if they don’t like it, they will eat almost nothing from it). So, as long as your rabbit is eating long strands of hay it’s fine.
Regarding your bale, having small not crunchy bits is not really typical of timothy – you never know what you will get from one bale / bag to another. You also can never know if your rabbit will love the hay or not. Sometimes, I have a perfectly fine good-smelling green-looking bag of hay and my rabbits won’t touch it (I bought 6 bags of oxbow’s orchard 2 months ago because it’s very good, very expensive and that my rabbits loved it when they got it a while ago and they won’t eat ANY of it).
When it comes to hay, rabbits can be terribly and mysteriously picky.

I totally agree with Aki, buns can be very picky about hay and it’s always a mystery to the human why the like one type over another.
Technically, you can feed a bun any type of grass hay. Timothy, orchard, oat , meadow, fescue, brome to name the most common types. Oat is particularly crunchy and often have some immature seed heads that the buns appreciate. Orchard is often tolerated by people who are allergic to Timothy.
You shouldn’t feed an adult bun legume hay (alfalfa, clover) except in tiny amounts as a treat, but all grass hays are fine. You can rotate types if you like, a variety gives the best nutritional coverage.