Hey Swizzle.
This is my kind of thread. I really enjoy grafting.
I found when grafting figs, you can begin doing so anytime after the last forecasted frost of spring for your area. This is just a good rule of thumb because you don't want any new growth being exposed to late frosts. I usually keep a close eye on the 7-10 day forecast to help me determine when would be best for me to start my first grafts.
I prefer not to start any grafts until spring (as apposed to indoors). Just because it's so much easier to let nature do the work for you.
I've done cleft, saddle, and bark/rind grafts right up into the early part of June. As long as you're seeing good growth in the rootstock you may be able to graft figs just about anytime up through June.
The main thing you have to worry about as summer approaches is the tender scion & new growth getting scorched by the sun. I've had this happen. For this reason I usually plant grafted trees in pots & then place them under a shade tree for a few weeks. Once they've hardened off a bit, then you can move into a sunny location if you want.
There are many great videos on youtube demonstrating a variety of graft types and procedures. I recommend you take the time to watch as many as you have patience for ;-) .
Hope this helps.
~Bill.