Online linear programming (OLP) has found broad applications in revenue management and resource allocation. State-of-the-art OLP algorithms achieve low regret by repeatedly solving linear programming (LP) subproblems that incorporate updated resource information. However, LP-based methods are computationally expensive and often inefficient for large-scale applications. By contrast, recent first-order OLP algorithms are more computationally efficient but typically suffer from weaker regret guarantees. To address these shortcomings, we propose a new algorithm that combines the strengths of LP-based and first-order OLP algorithms. Our algorithm re-solves the LP subproblems periodically at a predefined frequency $f$ and uses the latest dual prices to guide online decision-making. In parallel, a first-order method runs during each interval between LP re-solves and smooths resource consumption. Our algorithm achieves $\mathcal{O}(\log (T/f) + \sqrt{f})$ regret and delivers a "wait-less" online decision-making process that balances computational efficiency and regret guarantees. Extensive experiments demonstrate at least $10$-fold improvements in regret over first-order methods and $100$-fold improvements in runtime over LP-based methods.