I've popped out of an airway in the past where there has been a change of base level. I continued on the navaid as if I was still in the airway and then I just requested that ATC co-ordinate my return back into the airway when the base stepped down again.
If I recall correctly they requested an ETA, and I was reminded to "remain clear of controlled airspace". Approx 5-10 minutes before reaching the airway I was given a new SQUAWK code (classic sign that you are about to be passed to another frequency), I was passed to the agency controlling the airway who promptly cleared me in. Easy!
If my memory serves me correctly I was flying north from Exmor in the N864
Other worthy point of note was that the SQUAWK code given to use when I re-entered the airway was the same as was issued on departure from Exeter, this has happened on a number of occasions. I've always had first class service from ATC in terms of airways flying.
I would also re-iterate the icing points, be careful and make regular icing checks, both visually of the airframe, spinners, OAT etc and also watch for a reducing IAS. Always have a plan B whether it be a descent (obviously with terrain in mind) or en route diversion.
If you dont do it already consider making a note of the freezing level on departure as you climb through it. This will change as you travel through a changing airmass but I've often found it more accurate than the forecast levels. You will then know what level to request a descent too if you need to shed some ice.
I've descended before to shift some ice and requested a level based on the zero degree level I had noted down during my departure, and despite it being over an hour later it was still very similar and I flew along at the level that was just above 0 and cleared it up quickly.
Having this information will hopefully reduce your workload at what can already be quite busy by not having to request further descent or descending more than you need to.
Apologise if the icing lesson seems to come across as a lecture I only intend it as friendly advice
