The explosions for Orion that occur in the magnetosphere where the magnetic field lines lead back to earth is where fallout will come back down and be a problem.
We have already studied that reducing the fission component of any bomb and getting to higher fusion purity greatly reduces fallout and also a north pole launch reduces the fallout that returns to earth. Having a pile of conventional explosives for the first pulse also helps since the ground contact explosion is messier than the air bursts.
It would also seem best to send it up during a snow storm which would contain the fallout that coincides with a solar storm that flattens out the magnetosphere.
If you could not make the pure fusion bombs, which has not been done yet then another way to further reduce the radiation is for an unmanned high-G sprint start to a point outside the magnetosphere zone.
High-G Asteroid Interceptor
An unmanned Orion asteroid interceptor was designed. It would not need shock absorbers. Artillery arming, fusing, firing system for shells are regularly built to take 1000 Gs.
There was a three page paper: Nuclear explosive propelled Interceptor for deflecting objects on collision course with Earth. Johndale Solem, Los Alamos, proposed unmanned vehicle. No shock absorber or shielding. The pulse units were 25kg bombs of 2.5 kiloton yield.
Get to high velocities with only a few explosives and small shock absorbers or no shocks at all. Launch against a 100 meter chondritic asteroid coming at 25 km/sec. 1000 megatons if it hits. Launch when it is 15 million kilometers away and try to cause 10000km deflection. A minimal Orion weighing 3.3 tons with no warhead would do the job. 115 charges with a total of 288 kiloton yield. Launch to intercept in 5 hours. Ample time to launch a second if the first failed.
Sprinting out of the Magnetosphere
Notice the unmanned high acceleration configurations would reduce the number of charges to go through the atmosphere to about 1-3 charges. Instead of 200 charges to go to orbit with constant lower acceleration. Kick it hard with 3 or fewer 100G force acceleration charges. (charges would go off every half second for fast acceleration instead of 1.1 seconds for human safe acceleration).
It can head up at 100Gs. 980 m/s**2. So only 1-3 charges is enough to give escape velocity then coast. It is only a matter of containing the fallout from 1-3 low level charges. Plus 1-3 charges and that is it we have tens of thousands to millions of tons to start the space age.
Some of the Orion configurations were for 1000Gs of acceleration. At 100G's in 10 seconds it would be almost 50 kilometers up. 20 shots assuming one every 0.5 second. In 20 seconds it would be almost 200 kilometers up.
Some more charges could be used to slow the Orion for a rendezvous with human passengers and acceleration sensitive cargo. They could then fly anywhere in the solar system at a leisurely pace without concern about fallout.
Another aspect of the fast acceleration that is possible is that an unmanned Orion go from earth or earth orbit to Mars (decelerate at halfway) and get to Mars in under one day going at 100Gs if Mars and Earth are in the close approach. If the unmanned version was going at 1000Gs (which was a design that is possible), then Earth to Mars could be done in a few hours. At about 300Gs and you would be looking at a Mars Overnight package delivery.
Mars comes to about 50 million kilometers (36 million miles) of Earth on close orbital approach. This is just over three times the asteroid intercept scenario.
It is possible to get fit people to safely endure 30gs for one minute.
High acceleration compensation.
Equally important are some experimental protective techniques for increasing +Gz tolerance, currently under investigation. These methods include :
Pulsating G suits, synchronized to the electrocardiogram. This technique would provide a pulse superimposed on the systolic pulse, producing a higher systolic pressure at head level.
Optimization of physical fitness training procedures. This may allow a more forceful straining manoeuver with less fatigue.
Drugs to increase head level blood pressure on a short-term basis.
A liquid immersed and liquid breathing (special oxygen supplying liquid) person could sustain 15-80Gs. All air voids would need to have liquid. Ears, lungs etc... The liquid immersion and liquid breathing method has not been developed yet, but could work in theory and would increase the sustained acceleration tolerance to some higher level.