First I have used .270, .280 Rem and .25/06 cases to make 6.5 / 06 Ackley cases. I paid the least for the .270 Win cases and the most for .25/06 cases. I would recommend that the extra dollars for a hundred .25/06 cases is worthwhile. I had neck shoulder junction "donut" issues, even after case turning into the shoulder with the larger diameter necks. These were never as accurate as the .25/06 cases from the same (Nosler or Norma) manufacturer. I like to plink away at 200 to 300 yards at the centers from clay pigeons scavanged from the Trap Field at my Club Range. I use minimum load data from Sierra or Hodgdon and the 100 or 85 gr HPFB Varmint bullets from Sierra, Nosler and Hornady. Since I have a big game scope that stops at 12 power, this is about as far as I can get a bead with the rifle on two inch clay centers. With .25-06 cases, I still have to expand to 7mm (.270 is not enough) and then run them through a full length die to get a false shoulder to headspace on.
The alternative is to use a very long for your throat bullet. The discontinued 160 gr Sierra "Semi-Spitzer" or Hornady "Round Nose" are good choices with a case full of too slow of a powder like surplus H-870 or H-869. Don't start a forest fire with six feet of muzzle flame. See photo below: