In addition to what has already been recommended I would like to point to the 130gr Barnes TSX as well. In the single shot you have it can be loaded to a very respectable velocity and will give you some added range as well. In talking with Barnes a while back about some reduced .308 loads for my grandson they said that I could expect at least double caliber expansion down to around 1700fps.
Also as mentioned the 150gr Nosler BT, and Hornady SSor even the 155gr AMAX at normal velocities will all open up down to around the same velocity as the Barnes will. This will be for at least a double caliber expansion. So you would be talking a .308 bullet impact with it opening up to around or over .600" or so isn't bad as you will get the added sectional density which will give you more penetration. And yes the AMAX is primiarily designed as a competition bullet, but I have used them at the lower velocities for years in my daughters 6.5x55 with no problems. As long as the impact velocity is under 2500fps they expand reliably and penetrate very well in the lighter calibers.
Don't get in your mind that some of the 150gr bullets won't preform at the lower velocities, I shot a doe using 140gr Nosler BT@2250fps from a Contender chambered in 7x30 Waters and a 14" barrel. The shot was 283 yds using my laser, and entered behind the left shoulder and exited through the right shoulder pretty much destroying it and leaving close to a 2" exit hole. Everything from the initial hit to the exit was for the most part jello. I would have never thought it would have done that type of damage at that velocity had I not seen it first hand on that deer and several feral hogs afterwords. The best thing you can do is load some up and try them out in several different mediums or at least run them through a line of water filled milk jugs or wet newspaper. At least then you have some idea of what to expect.
Right now I am working on a few loads for a 30-30AI for my Contender which is chambered in a 14" barrel as well. Initial testing with a few powders has been real promising. I have a couple of loads which ar hitting the 2250fps mark with the 150gr Noslers. Using a ballistics program this would give me an effective range of 250yds for deer with the remaining velocity being just over 1750fps and energy being 1036fp. Compared to the 140gr load from the 7x30 which was nearly identical only with around 100fps more velocity. This is only due to the difference in B.C.'s between the two bullets. Even with this however, your still going to have the added bullet weight to help out.
Not knowing what your barrel length is with the Handi I would say that you should at least be abel to reach if not exceed the 2300fps mark using 150gr bullets which would enable you to reach out to 200 yds with confidence on deer sized game.
