Reid was moving well, showing no signs of adverse effects from the lingering knee problems that have keep him sidelined for 18 months.
But facing full blooded contact in a match situation is another matter, and both Hay and Reid will need to be certain of his readiness, rather than take unnecessary risks, as the player plots his return to action with West Ham.
Meanwhile Hay could select two debutants in his starting team for Friday's match (NZT).
Junior All Whites captain Joe Bell is likely to get the nod in central midfield, while Phoenix youngster Callum McCowatt is a strong option to be part of the front three.
Selection will cap a swift rise for both. Bell has made great strides since the Under-20 World Cup in Poland, and has probably edged ahead of Alex Rufer for the holding midfield spot.
McCowatt's star has been on the rise for a while, and the absence of Kosta Barbarouses has opened the door for the 20-year-old.
Reid and Michael Boxall will anchor the defence (with Tommy Smith or Bill Tuiloma to come in if Reid is not passed fit) with Storm Roux and Liberato Cacace the probable options outside them.
There are four players contesting the two central midfield spots (Ryan Thomas, Sarpreet Singh, Michael McGlinchey and Matt Ridenton) with Thomas and Singh the leading contenders.
Forwards Marco Rojas and Wood should round out the team.
The standout feature of Tuesday's session, aside from the bitterly cold conditions, fuelled by a bracing wind, was the emphasis on keeping possession.
Players, especially the wide ones, were berated for putting in hopeful, early crosses, when there was an option to recycle and all were encouraged to be brave in possession.
For some it will require an uncoiling of old habits especially as former coach Anthony Hudson encouraged a more direct approach, and it's not something that can be perfected in a few days, but players are positive.
"It's good to see that we are trying to play this way," said McGlinchey. "A lot of the young guys are good with the ball and it's very positive for this team."
The All Whites will have their final session on Wednesday morning, before a captain's run at Aviva Stadium in the afternoon, to experience the famous venue.