Well, not sure if I can explain this in writing very well, but one way of stepping your mast is to get another old (but straight) mast section and somehow attach a foot to the base to protect your deck. Something like some plywood with carpet that is permanently attached to the bottom of the mast (like another mast step casting, etc.). The mast should be taller than the height of your spreaders by several feet.
You then want to install a halyard in your new mast. Perhaps you have a masthead casting with sheaves, or you can attach a big enough block near the top with a hound or solid eye strap, etc. You also need to be able to four point the 2nd mast section so it is supported by itself (just like a real mast, but the deck attachment points will be different than your working mast (cleat, slotted aluminum toe rail, etc. - something sturdy). The halyard on the shorter mast could be wire and lead to a trailer winch mounted about 4' off the deck. This is not a single-handed job. You could use a regular winch too, but then you need someone to crank and someone to tail and someone to hold on to your working mast.
After you have all the components together, you, with the help of friends, stand up the short mast and place the foot next to the mast base for the working mast. Attach your shrouds (4-point it) so it stands alone.
Then attach the halyard to a sling or loop which is around the other mast - the loop being below the spreaders. Tie a line from the loop down to the gooseneck area, perhaps the winches, etc. and secure it so that when it stretches out, the loop will remain below the spreaders (not putting too much strain, if any, on the spreader bases). It can also fit under the lower shroud attachment points too.
With the halyard from the short mast attached to the loop beneath the spreaders (I believe the Morgan OI 28 was a single spreader rig. If a double spreader rig, then you must go to a point between the two spreaders and secure the loop in position with the line attached to the lower end of the working mast).
Essentially you will raise the working mast, which I assume is lying on the deck of your boat, with the shorter mast. Someone needs to hold on to the bottom of the working mast so that it stays put and also guides it towards the mast base where she is to be stepped upon. When you get the working mast upright and pretty parallel with the short mast, then you must four point it, so it stays in place.
You then remove the short mast and get it away from the boat so you can tune and finish rigging the working mast.
To be safe, you can take a halyard from the now stepped working mast and attach it to the short mast to lower it in a controlled manner while your helpers guide it down. Otherwise you need enough people to walk it down like a ladder, which could be a little scary.
I hope this makes sense. It is a bit of a job, but you can eliminate the need for a marina, but keep in mind, even though there is a cost, the marina will do it safely and if there is a problem, it is their problem.
But, that is one say a DIY can do it.
