Do your pedals have tabs on the back plate to help flip the pedal so you can stuff your foot into the clip? If not, you may want to consider pedals that have this feature. Not all tabs are alike, either:



The top pedal is an early Campy Record pedal, which has been widely copied by e.g. Gipiemme, Zeus, Olimpic, MKS, KKT, Shimano, and many others.

The bottom pedal is an even earlier design (1930s) by French track champ Marcel Berthet. The extra-large tab was designed explicitly for making pedal entry easier on fixed gear bikes. Unfortunately, they stopped production in the late 1980s, but after more than half a century of continuous production there are still many out there you might find. Make sure you get the right pedal thread when you find a pair! They were available in French (metric), English (ISO), and American (1/2" x 20tpi) threads. The Berthet pedals were also copied, but not as widely as the Campy pedals. Examples include the Sakae Ringyo SP-11:



The Mikashima (MKS) GR-9:



And more recently, the White Industries "Urban pedal:"



Oh, and what you've been calling "cages" are properly termed "toe clips."