Very good, on clearing the jam. To prevent further jams, stop using those Daisy pointed pellets (Seriously). Almost any other .177 pellet will feed (and shoot) better than those. Even the wadcutters (flat faced in simple terms). In fact, wadcutters would be more beneficial for rodent eradication as they hit harder and limit over-penetration. Don't take my word for it, just ask (or search)...
You can still buy Daisy's, just not pointed
You don't need a cleaning rod, but they are helpful for certain tasks... But for cleaning the bore of airguns, I prefer weed-eater line, Lol. Probably think I'm crazy... Taking a length of weed-eater line (any brand, preferably thinner) TWICE the length plus a few inches longer than your barrel... Find the middle and fold over making a loop, feed loop down the muzzle (dangerous end) of barrel until it exits the breech/chamber (where you load pellets). Then with a cleaning patch (cotton square) soaked with... cleaner (get to that soon)...insert into loop of weed eater line about mid way and then pull threw the barrel from muzzle end. Then repeat with clean/dry patches until they come out clean (or you're sick of pulling patches).
As for cleaner, some have used goo-gone. I prefer Ballistol (or Remoil if out) to clean and lube and protect, BUT... I'm not certain you need to do all this
Now for oiling, there's a few pivot points in the pump linkage that need oil and also the pump piston head... I've typed a lot, look to owners manual for locations
Next shopping trip get some wadcutters, domed and hollow points. The one that's most constantly accurate is what you should be using. Gun chooses ammo it likes, not the operator.
Good luck!