Being Stuck in a Class B - Some Ideas?

You can upgrade to Class-A CDL.
@Moose1958 knows the details on that.
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Where are you located; state and nearest city/town?
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Keep that hazmat endorsement because that gives you many more options. Even refrigerated companies haul hazmat.

 

If you have a class B CDL you are free to upgrade at any time. I am sorry to say that you get no class A credit for class B driving. As far as operating the vehicle the two are different. My son-in-law and were talking on the phone several days ago and he mentioned wanting to get a CDL. This is a man that is a Lt. Colonel in the US Army. I told him the same thing I will tell you. It's out there just go get it.

 

It is unclear to me what license you have. Part of your post suggests you have been doing a little class A work, but your boss keeps you mostly doing class B work.

If I were in your situation, I would get another job unless there are other personal reasons for remaining there. You are just wasting your time by staying and failing to get needed experience.

Can’t blame the boss, because he does have a business to run. Just accept that you aren’t a good fit and move on.

 

Yes, I agree. My comment was based on chinatown's comment mentioning me. After reading the OPs first comment again I too can see the OP is not clear on what class CDL they have.

 

I’m a bit confused here myself. First you mention your boss has you do some work ‘with a tractor’. Is there a trailer involved? If yes, you and your boss are in clear violation of regulations(if you are traveling on public roads). Now, I have been known to ‘bend and massage’ a few rules over the years, as most are outside the scope of the U.S. Constitution but that’s a different discussion for a different day.
Get as much time behind the wheel of a semi as you can. Yes, your boss is keeping you in a straight truck probably because you are good, and productive for his business to keep you there.
When you feel confident you can pass a road test to upgrade your cdl, then go for it. Contact your DMV, DPS, or whatever agency in your State conducts those tests, and go from there.

 

Did you ask your boss to assign you to Class A work only?

 

Yes, while many companies will look at your work seeing you are doing both, all your experience should be the same to a new employer. Even if your present employer says you are working driving a straight truck (it isn't class b work), you can counter this with the fact they put you in both types of trucks, straight and semi.

 

Those class B trucks are defective!
You back them to a door.
Looking in the mirrors.
Turn the wheel.
They go the wrong way!

 

If you have Class-A now, it's to easy to move to another job. Just put some applications in and you'll have a new job in a couple days. As in Post #8 you drove every truck the company had and it all counts as experience.