The Texas sun was still a promise of warmth as it crept over the Davis Mountains. I was sitting on the valley floor, nestled as comfortably as I could be into the scrub brush and cholla cactus, my binoculars glued to my eye sockets. High above, perched on the basalt columns, were several Barbary sheep, their shaggy manes blowing in the morning breeze.

Better known as aoudad, these North African transplants have taken up a comfortable residence in the arid Chihuahuan Desert. The difficulty in stalking aoudad in such rugged country, along with their heavy, sweeping horns, makes hunting them an affordable alternative to expensive trips for spiral-horned sheep. A D.I.Y. aoudad hunt will set you back about $2,000, and a guided hunt costs around $5,000—both a fraction of the price of a mountain sheep hunt, yet still wildly adventurous.

I spent the rest of that first day—and several that followed—picking my way up and down those stark, ragged remnants of ancient volcanoes, trying to crawl my way within range of a ram. By the end of the week, my muscles ached and my elbows and knees were prickled with cactus needles. I never did pull the trigger, but I promised to return, something made possible thanks to the money I’d saved.

Back in the 1930s and ’40s, a big-game hunter named Grancel Fitz became the first man to take all 28 Boone and Crockett species. He did it with a bolt-action .30/06. Nothing has changed. If you’d like to go everywhere and shoot everything, Mr. Fitz’s choice of caliber should be yours. In the hands of a good shot, there’s nothing the ’06 can’t drop. His rifle, however, was wood-stocked and iron-sighted, and there we part company. The rifle I’d select today is the Tikka T3 Lite Stainless. It’s a very light, all-stainless, synthetic-stocked gun that carries a guarantee of 1-inch MOA accuracy and will usually beat that by a good bit. It’s unstoppable, unbreakable, and rustproof given minimum maintenance. At under $700 (street price) you’ll have money left over for a superior scope. —D.E.P.

1. Stock

Synthetic. Neither snow nor rain nor heat nor gloom of night nor a horse rolling on it can disturb it.

2. Action

A glassy-smooth stainless action you don’t often encounter in rifles in this price bracket.

3. Trigger

Excellent pull and adjustable down to 2 pounds. Leave it at 3.

4. Magazine

A detachable box, so you can load and unload in a hurry if necessary.

5. Scope and binoculars

The Meopta MeoStar R2 1.7–10x42 RD ($1,450; has the perfect power range for all big-game hunting. The optics are first class, take your best compact binoculars, the adjustments are fine, and the illuminated reticle is a big help. It ain’t cheap. So what?

6. Barrel

Stainless steel. Hunt in the rain for a week. Or two. The rifle will not care.

7. Sling

The Murray Leather A-1 Quick Set Rifle Sling ($75; ) is the best I know of—dead simple, and given a dose of vegetable oil once in a while, it will last forever.

Mounts

Get Tikka rings and its Picatinny rail, which offers huge latitude in fore and aft adjustment.

Weight

About 71⁄2 pounds with scope. Not too much to carry, and enough to hold steady when your heart is going pitter-patter.