GPSMAP 66i external antenna...

I think not possible because the 66i has no port for an external antenna like the 64s has. 

For inductively coupled search for a 'GPS re-radiator'.  Howerver they require their own power supply and the one I have and others are not really designed for bush walking.  Example: https://www.ja-gps.com.au/GPS-Express/re-radiating-gps-antenna.  I've used one like this for about 15 years for easier setting up of GPS's indoors.  The one on GPS City says longer receiving cables are an option.  I contacted https://trackme.nz/ who are based in Christchurch and already have an answer which negates the above and I should have thought of:

"that [GPS re-radiator] won't work other than to give a GPS location. It won't pass the Iridium signal via this, there are solutions for this but they are pricey. The ability to transmit through canopy is linked to the density and frequency of cover, if you are moving under dense bush very hard to get the transmission out, if you are stationary only the hardiest of canopy will thwart your signal, it will eventually find a way out."

How should you connect the antenna to the 66i since the device does not have an external antenna connection?

In the case of a re-radiator, no physical connection to the GPS is required.  The device transmits the GPS signals from the short 'antenna' at the end of the indoor cable. Thus it's easy to get GPS satellite signals to as many GPS units as you have within a metre or two to the indoor antenna.

Ah, thanks.  Didn’t know that. 

I am unclear as to whether you are asking about GPS signal strength or iR communication. All of the answers in this thread are regarding GPS signal strength. I am not aware of ANYTHING which would help with iR communication.

Thanks for replying,  per my original post:

"Hence, to solve the no InReach messaging possible under the tree canopy issue:"

So as I said in the post, iR messaging.

Sorry if that got lost in the verbiage somewhere!

Cheers. 

It's always hard to distinguish. The iR devices will warn you when you try to send without a GPS fix. You can choose to send anyway, but the message will not have an associated location. (I believe track points for which no fix is available are dropped, though. See following.)

My original comment with regard to iR messaging itself still stands. AFAIK, there is no solution other than the get to a place with a clearer sky view. This is one of the reasons why I always recommend iR tracking when in the back country. If you end up stuck in a location with no available iR messaging, at least SAR can work with your last known position/track point. Also, messages/track points which cannot be transmitted immediately are queued for later transmission. When you eventually emerge from cover, they will go out. (There is a limit to the number of track points which will be queued, however.)