Third-placed Boro are five points off pole position and welcome Little Oakley to the Lawtech Stadium on Saturday, then visit second-placed West Essex on Tuesday.
And Spinks wants to remain among the front-runners for promotion before winter sets in.
“We want to try and keep up with Woodford before we play them, gain some points on them, that’s the plan,” he said.
“We’ve got three or four games coming up that we should be winning. I don’t like to say that, but we hope to win them.
“We’re all playing each other at the top, there will be some tough games between each other.
“But we’re barely 10 games into it and it can all change again over the winter.”
Boro had to settle for a share of the spoils at Halstead Town last weekend, after conceding a late goal to the Humbugs.
Hassan Nalbant’s ninth goal of the season gave them an early lead at Rosemary Lane, two months on from a 4-1 FA Cup win at the same ground.
But the home side had the better of things for the most part and forced young goalkeeper Jack Dixson into a string of saves.
The Dagenham & Redbridge loanee was eventually beaten two minutes from time, though, by Joel Older and Boro saw Jacob Wiggins dismissed for a second bookable offence in stoppage time.
“They were two half-late tackles so no complaints,” added Spinks.
“The first 15 minutes was the best we’ve been this season, we were awesome. But the other 75 was probably the worst we’ve been.
“They deserved a point in the end, I can’t complain with the result, but I can complain about the performance.
“If we’re disappointed with only getting a point from a tough side away from home, that’s a good sign.
“I wasn’t best pleased but I’m looking at the bigger picture.”
Spinks was due to watch Dixson, Henry Hart and the fit-again Finlay Dorrell in FA Youth Cup action as Barking met Dagenham & Redbridge in midweek.
And he is looking for more support for leading scorer Nalbant in the coming weeks.
“Jack did ok on Saturday and we’ve got Jake [Anderson] coming back now so there is more competition for places,” he added.
“Finlay will be back on Saturday but we’re still without Junior [Luke]. He was our best player for the first eight or nine games of the season.
“I don’t really care who scores. Hassan has been on fire, he’s been excellent. But we need more goals from others.”
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