If you are fully laid off, which means not working AT ALL for your original employer, you are not receiving partial unemployment. You are receiving a full unemployment amount each week....if they are going to call you back in the spring or when the weather breaks, or they get in more work, etc. you are "job attached," which means you do not have to look for and accept other work to draw benefits.
Partial unemployment would be if you got some hours on your regular job, but not enough to make more than the amount you could draw a week in gross wages, or that you were off for a couple of weeks for vacation, going right back to work, maybe receiving vacation pay. This is usually filed by the employer. But if you are going to be off more than six or eight weeks in a row, you usually have to file for regular benefits.
If you take another part time job in the winter, you will need to report all your gross earnings from that job, weekly. If the employer is doing your unemployment using partial forms, you still need to tell the unemployment office how much you are making each in gross wages from the other job. If the amount you make before take outs is less than the amount you could've drawn in weekly benefits, even one dollar less, then you will get a check for part of your benefits for that week. Usually by doing this, you can pick up another $75-100 a week, depending on the state.
The amount you do not get is not lost, it is added onto the balance of your claim, which means you might have a few more weeks to draw than otherwise.
If you were to quit or get fired from this part-time job during this time, it would also cause a problem with your eligiblity for regular benefits, as each week you have to answer, "Did you quit a job, get fired from a job or refuse a job?" in order to certify for that week.
But if you take a part time job that pays more than your weekly benefit amount, you will simply report what you made the first week you have it, which will be more than the weekly amount of U.I., and this will stop your claim. You can't draw any of it while making more than the WBA. (It has certainly been tried, just not reporting the wages, or being paid cash..I very much advise against it, as it is fraud and will be caught by the system or reported by an ex-wife! Kidding!)