And the company does not "give unemployment" anyway. If they have told you, which they have, that you are being let go due to their not having enough work for you, and that they have only part time hours for you during the next two or three weeks, then it would be fine for you to continue to work those part time hours. But after the first Sunday through Saturday week that you only get part time, FILE YOUR CLAIM FOR UNEMPLOYMENT BENEFITS IMMEDIATELY. State what your employer has told you. State clearly that you are still working, but you are working all the hours that your employer has for you, and you only made (you figure this, regardless of how it is calculated or paid by your employer) xxx dollars in gross wages, from Sunday, April 24-through Saturday, May 1. If you don't have the just exact figure, you can round it up a little bit, just say $12 an hour for 10 hours, = $120 in gross wages. This will set you up for partial unemployment benefits. This will be your waiting week on the claim.
If you work that second week, then for the second week, after you've filed the claim and certified as you are supposed to for the second week, they'll pay you part of your unemployment benefits in addition to the money you will eventually be paid for working at the job that week. Just be sure and report it honestly as best you can.
After that, of course, you'll be certifying for weeks when you had no work at all, and you will receive the full amount of unemployment in your weekly benefit amount.
Just be sure you tell the unemployment system exactly what your employer told you when you file the claim. You do not need some sort of official statement or paper separation notice, though it would be nice if they gave you one. But even if they do, usually the company is still contacted and asked for the reason you no longer work there. If they lied, and said you had been fired for some cause, then that would be investigated, but their word on this matter does not have any more weight than yours. A lot of employers try to keep people from drawing unemployment by firing them for made up reasons or trying to make the job so bad they will quit (usually this keeps you from getting unemployment benefits) so your employer, by telling you honestly that you are laid off because they do not need you any more, is treating you the best way they can under the circumstances.
Unemployment insurance, if you are approved, even if the employer does not contest it, will take several weeks to get started, and will be usually quite a bit less money than you could make working. It lasts only a maximum of 26 weeks, during which you must do everything they tell you as far as participating in their job searching activities, you must be making weekly certifications that you are able, available and actively seeking other work, and it ends, regardless of whether or not you've found another job or what your circumstances are. So get signed up on it and start looking hard for another job.