need advice for a stealthy chipmunk eliminator

If you went to the springer forum and asked for a recommendation for what you are trying to do, one of the top recommendations would be the R7/HW30 (same gun). It is possible you have a lemon, but probably not. If you have access to a chronograph, velocity should be somewhere around 650 fps with an 8 grain pellet I believe. Someone else might be able to provide a better number for velocity. If you don't have a chrony, it should shoot through both sides of an ordinary tin can at 15 yards with round nose pellets.

Servicing airguns is pretty much a DIY thing, although sometimes there will be a local guy who will take it on. It isn't hard, but you do have to know what you are doing. Fortunately there are lots of videos on youtube, and forum guys willing to give free advice. If the gun does seem to be making reasonable power, and it probably is, you might be able to work some springer rated grease through the cocking slot and onto the spring and inner walls of the receiver until you can take it apart this winter for a proper servicing. But remember less is more! You only want a light coating of grease on the internals. Vortek and ARH sell lubricants. I will never put chamber oil in a gun again as it has caused me more harm than good.

The scope and mount recommended by Yo are top notch, and the scope is a great deal at $299. Just buy them, they will be used on whatever gun you end up using. I assume the gun has an aperture sight at the rear? With experience you can become proficient with iron sights, but a scope is always going to be an advantage, especially on game. The scope will also add some weight, which aids accuracy in a springer.

Pointed pellets have always been the least accurate for me. For shooting at any kind of distance I prefer the round nose field target style pellets, either JSB or H&N at around 8 grains for that power level. Although Yo keeps recommending the Premier HPs, so give them a try as well.

And lastly, springers may or may or may not respond well to standard bench rest techniques, usually not. Try one bag in front of the trigger guard, just in front of the balance point. At 10-15 yards groups should be 1/2"-3/4" to start, and can be much better than that with practice.
 
Yes, if the gun has been in a garage for 25 years I would be concerned about rust, maybe even in the bore. On a break barrel I use a standard cleaning rod and jag, but some prefer a pull through. I just run some oiled patches through followed by a dry patch or two. If patches come out rust colored, it is time to attempt a rescue with JB bore paste.
 
The R7 doesn't recoil hard enough to "destroy" scopes. The Hawke Vantage is rated by Hawke themselves for spring guns under 12 fpe and the R7 is. While I have more than a few, I'm a bigger fan of the Airmax line, which yes, were designed and built for use on spring piston air rifles and I own 7, 5 of those on springers one of which just happens to be an R7 .20 caliber that's only about 9 years newer than yours and has an Airmax 3-9x40 on it.

An R7 also most definitely has no need for a 6 screw base, 1 piece scope mount. Nothing personal Yo, but while Sportsmatch is nice, that scope mount is overkill for an R7. They only shoot 6-8 fpe and I'm one and I've seen any number of other owners who use 2 piece ring sets to mount their scopes. Mine has a set of reversed Hawke Reach Forward Match mounts (I like my scopes to sit back farther) although a simple set of the Hawke medium Match mounts or a simpler 2 or 3 screw base 1 piece mount would also work just fine.

By all means follow all the advice for cleaning the barrel and checking the screws and a simple way of doing a lube tune until you can send it to a qualified air gun tuner for a going over would be Air Venturi's Tune in a Tube. No, it's not a permanent solution, but it is a quick fix for lubing a rifle that likely needs it. Link to Rick Eustler's how to video to at least consider it: https://www.airgunweb.com/tune-in-a-tube-review/ and as already suggested, try the H&N and JSB pellets and practice, practice, practice.

Mine in .20 (obligatory pic below...lol) has easily taken squirrels with headshots out to 25 yards,
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so chipmunks should be just as easy and I've seen plenty of owners of the HW30/R7 rifles say they shoot both regularly. 


 
Ok, so I'm going to order a scope and mounts for the R7 and see if that solves most of the problem.

If not, and if I'm itching to buy a new air rifle, I'm leaning towards staying with a springer mostly because that's what I'm used to. After some time reading / viewing online, I'm thinking that a Weirauch 97k might be a nice upgrade. I like the idea of the fixed barrel.

Therefore, would the Bushnell 5-15x40 work with the Weihrauch 97k, too? I'm reading that some scopes are rated for airguns of different strengths and some are not.

Same for the scope mount. Would the one that Yo linked to work on the 97k, too, or is there a less expensive, but just as good, option that I should look at?
 
BTW in order to MATCH the OPTICAL CLARITY of the $299 ($499 REGULAR PRICE) JAPAN Bushnell TAC 5-15x40 AO ON SALE AT SWFA.COM you will have to spend $699 for a SWFA SS 3-15x42. ABSOLUTELY NOTHING in the UNDER $700 PRICE RANGE WILL MATCH THE CLARITY OF THAT BUSHNELL TAC 5-15x40AO EXCEPT THE $229.99 AIMSPORTS ALPHA6 4.5-27x50 and 2.5-15x50 version but I HIGHLY DOUBT THEY WILL HOLD UP WELL ON SPRINGERS SO BEST NOT TO EVEN TRY. AND in order to MATCH the CLARITY OF THE AIMSPORTS ALPHA6 SCOPES YOU NEED TO SPEND $800 AND UP, AND IN ORDER TO BEAT THE AIMSPORTS ALPHA6s IN OPTICAL CLARITY YOU WILL NEED TO SPEND $1000 AND UP.